# Calculus

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

I recommend this book: A Course of Modern Analysis by Whittaker and Watson. You may also find this book at Google Books. This book is a hundred years old and is considered the classic calculus book.

## Derivatives

### Definition of Derivative

${\displaystyle f'(x)=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$, provided the limit exists.

For the following problems, find the derivative using the definition of the derivative.

solution ${\displaystyle f(x)=x\,}$

solution ${\displaystyle f(x)=2x^{3}\,}$

solution ${\displaystyle f(x)={\sqrt {x}}\,}$

solution ${\displaystyle f(x)={\frac {1}{x}}\,}$

solution ${\displaystyle f(x)=\sin x\,}$

solution ${\displaystyle h(x)=f(x)+g(x)\,}$

### Power Rule

${\displaystyle {\frac {d}{dx}}(x^{n})=nx^{n-1}\,}$

For the following problems, compute the derivative of ${\displaystyle y=f(x)\,}$ with respect to ${\displaystyle x\,}$

solution ${\displaystyle f(x)=150\pi \,}$

solution ${\displaystyle f(x)=7x^{4}\,}$

solution ${\displaystyle f(x)=27x^{2}+{\frac {12}{\pi }}x+e^{e^{\pi }}\,}$

solution ${\displaystyle f(x)=8x^{5}+4x^{4}+6x^{3}+x+7\,}$

solution ${\displaystyle f(x)={\frac {1}{\sqrt[{4}]{x}}}\,}$

solution ${\displaystyle f(x)=7x^{-4}+12x^{-2}-x^{\frac {1}{2}}+6x+\pi x^{\frac {3}{5}}+x^{-{\frac {3}{4}}}\,}$

solution Derive the power rule for positive integer powers from the definition of the derivative (Hint: Use the Binomial Expansion)

### Product Rule

${\displaystyle {\frac {d}{dx}}(ab)=ab'+a'b\,}$

For the following problems, compute the derivative of ${\displaystyle y=f(x)\,}$ with respect to ${\displaystyle x\,}$

solution ${\displaystyle f(x)=x\sin x\,}$

solution ${\displaystyle f(x)=x^{3}\cos x\,}$

solution ${\displaystyle f(x)=\tan x\sec x\,}$

solution ${\displaystyle f(x)=2x\sin(2x)\,}$

solution ${\displaystyle f(x)=(x)(x+7)(x-12)\,}$

solution Derive the Product Rule using the definition of the derivative

### Quotient Rule

${\displaystyle \left({\frac {a}{b}}\right)'={\frac {ba'-ab'}{b^{2}}}\,}$

For the following problems, compute the derivative of ${\displaystyle y=f(x)\,}$ with respect to x

solution ${\displaystyle f(x)={\frac {x}{x+1}}\,}$

solution ${\displaystyle f(x)={\frac {x^{2}}{\sin x}}\,}$

solution ${\displaystyle f(x)={\frac {\sin ^{2}x}{x^{3}}}\,}$

solution ${\displaystyle f(x)={\frac {x\sin x}{e^{x}}}\,}$

solution ${\displaystyle f(x)={\frac {x+7}{(x-6)(x+2)}}\,}$

solution Derive the Quotient Rule formula. (Hint: Use the Product Rule).

### Generalized Power Rule

${\displaystyle {\frac {d}{dx}}(f(x))^{n}=n(f(x))^{n-1}f'(x)\,}$

For the following problems, compute the derivative of ${\displaystyle y=f(x)\,}$ with respect to x

solution ${\displaystyle f(x)=(x^{3}+7x)^{3}\,}$

solution ${\displaystyle f(x)=\sin ^{4}(x)\,}$

solution ${\displaystyle f(x)=\left(\ln x\right)^{-4}\,}$

solution ${\displaystyle f(x)=\tan ^{2}x+8(x^{2}+4x+3)^{9}+\sec ^{3}x\,}$

solution ${\displaystyle f(x)=\left({\frac {5x}{7x+9}}\right)^{3}\,}$

### Chain Rule

${\displaystyle {\frac {d}{dx}}f(g(x))=f'(g(x))g'(x)\,}$

For the following problems, compute the derivative of ${\displaystyle y=f(x)\,}$ with respect to x

solution ${\displaystyle f(x)=\ln(7x^{2}e^{x}\sin x)\,}$

solution ${\displaystyle f(x)=\sin ^{2}(7x+5)+\cos ^{2}(7x+5)\,}$

solution ${\displaystyle f(x)=6e^{3x}\tan(5x)\,}$

solution ${\displaystyle f(x)=\ln(\sin(e^{x}))\,}$

solution ${\displaystyle f(x)=\sin(\cos(\tan x))\,}$

solution ${\displaystyle f(x)=e^{x^{2}}\sin(14x)-\cos(e^{x})\,}$

solution ${\displaystyle f(x)={\frac {\frac {\sin(5x)}{(x^{2}+1)^{2}}}{\cos ^{3}(3x)-1}}\,}$

solution ${\displaystyle \left(f(g(x))\right)'=f'(g(x))g'(x)}$

solution ${\displaystyle {\sqrt[{}]{x+{\sqrt[{}]{x+{\sqrt[{}]{x}}}}}}}$

solution ${\displaystyle f(x)=x^{2}{\sqrt {9-x^{2}}}\,}$

### Implicit Differentiation

For the following problems, compute the derivative of ${\displaystyle y=f(x)\,}$ with respect to x

solution ${\displaystyle \sin(xy)=x\,}$

solution ${\displaystyle x+xy+x^{2}+xy^{2}=0\,}$

solution ${\displaystyle y=x(y+1)\,}$

solution ${\displaystyle yx=x^{y}\,}$ where ${\displaystyle y\,}$ is a function of ${\displaystyle x\,}$

### Logarithmic Differentiation

For the following problems, compute the derivative of ${\displaystyle y=f(x)\,}$ with respect to x

solution ${\displaystyle f(x)=4^{\sin x}\,}$

solution ${\displaystyle f(x)=x^{x}\,}$

solution ${\displaystyle f(x)=x^{x^{{\cdot }^{{\cdot }^{\cdot }}}}\,}$

solution ${\displaystyle f(x)=g(x)^{h(x)}\,}$ for any functions ${\displaystyle g(x)\,}$ and ${\displaystyle h(x)\,}$ where ${\displaystyle g(x)\neq 0\,}$

### Second Fundamental Theorem of Calculus

${\displaystyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x)\,}$

For the following problems, compute the derivative of ${\displaystyle y=f(x)\,}$ with respect to x

solution ${\displaystyle f(x)=\int _{0}^{x}e^{t}\,dt\,}$

solution ${\displaystyle f(x)=\int _{5}^{x}27t^{t}\sin(t-1)\,dt\,}$

solution ${\displaystyle f(x)=\int _{4}^{x^{2}}\sin(e^{t})\,dt\,}$

solution ${\displaystyle f(x)=\int _{x^{2}}^{3x^{4}}\cos(t)\,dt\,}$

solution Give a proof of the theorem

## Applications of Derivatives

### Slope of the Tangent Line

solution Find the slope of the tangent line to the graph ${\displaystyle f(x)=6x}$ when ${\displaystyle x=3}$.

solution Find the slope of the tangent line to the graph ${\displaystyle f(x)=\sin ^{2}x}$ when ${\displaystyle x=\pi }$.

solution Find the slope of the tangent line to the graph ${\displaystyle f(x)=\cos x+x^{5}}$ when ${\displaystyle x={\frac {\pi }{2}}}$.

solution Find the equation of the tangent line to the graph ${\displaystyle f(x)=xe^{x}+x+5}$ when ${\displaystyle x=0}$.

solution Find the slope of the tangent line to the graph ${\displaystyle f(x)=x^{2}}$ when ${\displaystyle x=0,1,2,3,4,5,6}$.

solution Find the slope of the tangent line to the graph ${\displaystyle x^{2}+y^{2}=9}$ at the point ${\displaystyle (0,-3)}$.

solution Find the equation of the tangent line to the graph ${\displaystyle xy=y^{2}x^{2}+y+x+2}$ at the point ${\displaystyle (0,-2)}$.

### Extrema (Maxima and Minima)

solution Find the absolute minimum and maximum on ${\displaystyle [-1,5]}$ of the function ${\displaystyle f(x)=(1-x)e^{x}}$.

solution Find the absolute minimum and maximum on ${\displaystyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}$ of the function ${\displaystyle f(x)=\sin(x^{2})}$.

solution Find all local minima and maxima of the function ${\displaystyle f(x)=x^{2}}$.

solution Find all local minima and maxima of the function ${\displaystyle f(x)={\frac {x^{2}-1}{x}}}$.

solution If a farmer wants to put up a fence along a river, so that only 3 sides need to be fenced in, what is the largest area he can fence with 100 feet of fence?

solution If a fence is to be made with three pens, the three connected side-by-side, find the dimensions which give the largest total area if 200 feet of fence are to be used.

solution Find the local minima and maxima of the function ${\displaystyle f(x)={\sqrt {x}}}$.

### Related Rates

solution A clock face has a 12 inch diameter, a 5.5-inch second hand, a 5 inch minute hand and a 3 inch hour hand. When it is exactly 3:30, calculate the rate at which the distance between the tip of any one of these hands and the 9 o'clock position is changing.

solution A spherical container of ${\displaystyle r}$ meters is being filled with a liquid at a rate of ${\displaystyle \rho \,{\rm {m}}^{3}/{\rm {min}}}$. At what rate is the height of the liquid in the container changing with respect to time?

## Integrals

### Integration by Substitution

solution ${\displaystyle \int {\frac {2x}{x^{2}+1}}\,dx\,}$

solution ${\displaystyle \int x^{2}\sin x^{3}\,dx\,}$

solution ${\displaystyle \int \cot x\,dx\,}$

solution ${\displaystyle \int \tan x\sec ^{2}x\,dx\,}$

solution ${\displaystyle \int {\frac {\ln x^{2}}{x}}\,dx\,}$

solution ${\displaystyle \int _{0}^{3}{\frac {2x+1}{x^{2}+x+7}}\,dx\,}$

solution ${\displaystyle \int _{1}^{e^{\pi }}{\frac {\sin(\ln x)}{x}}\,dx\,}$

solution ${\displaystyle \int {\frac {x}{\sqrt {4+x^{2}}}}\,dx\,}$

solution ${\displaystyle \int {\frac {x}{1-x^{2}}}\,dx\,}$

### Integration by Parts

${\displaystyle \int u\,dv=uv-\int v\,du\,}$

solution ${\displaystyle \int \ln x\,dx\,}$

solution ${\displaystyle \int x\sin(x)\,dx\,}$

solution ${\displaystyle \int \arctan(2x)\,dx\,}$

solution ${\displaystyle \int e^{x}\sin x\,dx\,}$

solution ${\displaystyle \int x^{2}e^{x}\,dx\,}$

solution ${\displaystyle \int (x^{3}+1)\cos x\,dx\,}$

solution ${\displaystyle \int x^{4}\sin x\,dx\,}$

solution ${\displaystyle \int {\frac {\ln x}{x}}\,dx\,}$

### Trigonometric Integrals

solution ${\displaystyle \int \sin ^{2}(x)\,dx\,}$

solution ${\displaystyle \int \tan ^{2}(x)\,dx\,}$

solution ${\displaystyle \int \sin ^{5}x\cos ^{5}x\,dx\,}$

solution ${\displaystyle \int \sin ^{2}x\cos ^{3}x\,dx\,}$

solution ${\displaystyle \int \sin ^{2}x\cos ^{2}x\,dx\,}$

solution ${\displaystyle \int \tan ^{2}x\sec ^{4}x\,dx\,}$

solution ${\displaystyle \int \tan ^{3}x\sec ^{3}x\,dx\,}$

### Trigonometric Substitution

solution ${\displaystyle \int {\frac {x\,dx}{\sqrt {3-2x-x^{2}}}}}$

solution ${\displaystyle \int \operatorname {arcsec} x\,dx\,}$

solution ${\displaystyle \int {\frac {1}{\sqrt {4x-x^{2}}}}\,dx\,}$

solution ${\displaystyle \int x\arcsin x\,dx\,}$

solution ${\displaystyle \int {\frac {x}{1-x^{2}}}\,dx\,}$

solution ${\displaystyle \int {\frac {1}{(x^{2}+1)^{\frac {3}{2}}}}\,dx\,}$

solution ${\displaystyle \int {\frac {\sqrt {x^{2}-1}}{x}}\,dx\,}$

solution ${\displaystyle \int {\frac {x}{\sqrt {4+x^{2}}}}\,dx\,}$

### Partial Fractions

solution ${\displaystyle \int {\frac {1}{x^{2}-1}}\,dx\,}$

solution ${\displaystyle \int {\frac {1}{(x+1)(x+2)(x+3)}}\,dx\,}$

solution ${\displaystyle \int {\frac {x!}{(x+n)!}}\,dx\,}$

solution ${\displaystyle \int {\frac {x}{1-x^{2}}}\,dx\,}$

solution ${\displaystyle \int {\frac {dx}{\sin ^{2}(x)-\cos ^{2}(x)}}}$

### Special Functions

solution A ball is thrown up into the air from the ground. How high will it go?

solution Let ${\displaystyle f\,}$ be a continuous function for ${\displaystyle x\geq a\,}$.
Show that ${\displaystyle \int _{a}^{x}\int _{a}^{s}f(y)\,dy\,ds=\int _{a}^{x}f(y)(x-y)\,dy\,}$

solution Evaluate ${\displaystyle \int e^{x^{2}}\,dx\,}$

solution ${\displaystyle \int _{0}^{\infty }3^{-4z^{2}}dz\,}$

solution ${\displaystyle \int _{0}^{\infty }x^{m}e^{-ax^{n}}dx\,}$

solution ${\displaystyle \int _{0}^{\infty }x^{4}e^{-x^{3}}\,dx\,}$

solution ${\displaystyle \int _{0}^{\infty }e^{-pt}{\sqrt {t}}\,dt\,}$

## Applications of Integration

### Area Under the Curve

solution Find the area under the curve ${\displaystyle f(x)=x^{2}}$ on the interval ${\displaystyle [-1,1]}$.

solution Find the total area between the curve ${\displaystyle f(x)=\cos x}$ and the x-axis on the interval ${\displaystyle [0,2\pi ]}$.

solution Find the area under the curve ${\displaystyle f(x)=x^{3}+4x^{2}-7x+8}$ on the interval ${\displaystyle [0,1]}$.

solution Using calculus, find the formula for the area of a rectangle.

solution Derive the formula for the area of a circle with arbitrary radius r.

### Volume

#### Disc Method

The disc method is a special case of the method of cross-sectional areas to find volumes, using a circle as the cross-section.

To find the volume of a solid of revolution, using the disc method, use one of the two formulas below. ${\displaystyle R(x)}$ is the radius of the cross-sectional circle at any point.

If you have a horizontal axis of revolution

${\displaystyle V=\pi \int _{a}^{b}[R(x)]^{2}\,dx}$

If you have a vertical axis of revolution

${\displaystyle V=\pi \int _{c}^{d}[R(y)]^{2}\,dy}$

solution Find the volume of the solid generated by revolving the line ${\displaystyle y=x}$ around the x-axis, where ${\displaystyle 0\leq x\leq 4}$.

solution Find the volume of the solid generated by revolving the region bounded by ${\displaystyle y=x^{2}}$ and ${\displaystyle y=4x-x^{2}}$ around the x-axis.

solution Find the volume of the solid generated by revolving the region bounded by ${\displaystyle y=x^{2}}$ and ${\displaystyle y=x^{3}}$ around the x-axis.

solution Find the volume of the solid generated by revolving the region bounded by ${\displaystyle y=x^{2}}$ and ${\displaystyle y=x^{3}}$ around the y-axis.

solution Find the volume of the solid generated by revolving the region bounded by ${\displaystyle y={\sqrt {x}}}$, the x-axis and the line ${\displaystyle x=4}$ around the x-axis.

#### Cross-sectional Areas

solution Find the volume, on the interval ${\displaystyle 0\leq x\leq 3}$, of a 3-D object whose cross-section at any given point is a square with side length ${\displaystyle x^{2}-9x}$.

solution Find the volume, on the interval ${\displaystyle 0, of a 3-D object whose cross-section at any given point is an equilateral triangle with side length ${\displaystyle \sin {\frac {x}{2}}}$.

solution Find the volume of a cylinder with radius 3 and height 10.

### Arc Length

${\displaystyle L=\int _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx=\int _{p}^{q}{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\,dt}$

solution Calculate the arc length of the curve ${\displaystyle y=x^{2}}$ from ${\displaystyle x=0}$ to ${\displaystyle x=4}$.

solution Determine the arc length of the curve given by x = t cos t , y = t sin t from t=0

solution Calculate the arc length of y=cosh x from x=0 to x

### Mean Value Theorem

${\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}.}$

solution Find the average value of the function ${\displaystyle f(x)=e^{2x}}$ on the interval ${\displaystyle [0,4]}$.

solution Find the average value of the function ${\displaystyle 2sec^{2}x}$ on the interval ${\displaystyle [0,{\frac {\pi }{4}}]}$.

solution Find the average speed of a car, starting at time 0, if it drives for 5 hours and its speed at time t (in hours) is given by ${\displaystyle s(t)=5t^{2}+7t+e^{t}}$.

solution Find the average value of the function ${\displaystyle \sin ^{6}tcos^{3}t}$ on the interval ${\displaystyle [0,{\frac {\pi }{2}}]}$.

solution Deduce the Mean Value Theorem from Rolle's Theorem.

## Series of Real Numbers

### nth Term Test

If the series ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ converges, then ${\displaystyle \lim _{n\rightarrow \infty }a_{n}=0}$.

Note: This leads to a test for divergence for those series whose terms do not go to ${\displaystyle 0}$ but it does not tell us if any series converges.

solution Discuss the convergence or divergence of the series with terms ${\displaystyle \{-1,1,-1,1,-1,1,...\}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}}$ and ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=0}^{\infty }\sin n}$.

### Telescopic Series

If ${\displaystyle \{a_{k}\}}$ is a convergent real sequence, then ${\displaystyle \sum _{k=1}^{\infty }(a_{k}-a_{k+1})=a_{1}-\lim _{k\rightarrow \infty }a_{k}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=1}^{\infty }\left({\frac {1}{k+7}}-{\frac {1}{k+8}}\right)}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=0}^{\infty }{\frac {2}{(k+1)(k+3)}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=2}^{\infty }\ln \left({\frac {k(k+2)}{(k+1)^{2}}}\right)}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=4}^{\infty }\left({\frac {1}{k}}-{\frac {1}{k+2}}\right)}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=0}^{\infty }\left(e^{-k}-e^{-(k+1)}\right)}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=4}^{\infty }\left(e^{-k+3}-e^{-k+1}\right)}$.

### Geometric Series

The series ${\displaystyle \sum _{n=0}^{\infty }r^{n}}$ converges if ${\displaystyle -1 and, moreover, it converges to ${\displaystyle {\frac {1}{1-r}}}$. For any other value of r, the series diverges. More generally, the finite series, ${\displaystyle \sum _{n=a}^{b}r^{n}={\frac {r^{a}-r^{b+1}}{1-r}}}$ for any value of ${\displaystyle r}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=0}^{\infty }\left({\frac {1}{4}}\right)^{n}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=3}^{\infty }\left({\frac {2}{3}}\right)^{n}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=0}^{\infty }1.5^{n}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=0}^{\infty }\left({\frac {1}{6}}\right)^{n+2}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=10}^{\infty }\left({\frac {3}{5}}\right)^{n}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=7}^{\infty }\left(-{\frac {1}{2}}\right)^{n}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=7}^{\infty }\left[\left(-{\frac {4}{7}}\right)^{n+3}+\left({\frac {1}{3}}\right)^{n-2}\right]\,}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=0}^{\infty }{\frac {3^{n+1}}{7^{n}}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=0}^{\infty }\left({\frac {1}{3}}\right)^{2n}}$.

### Integral Test

If the function ${\displaystyle f}$ is positive, continuous, and decreasing for ${\displaystyle x\geq 1}$, then

${\displaystyle \sum _{n=1}^{\infty }f(n)\,}$ and ${\displaystyle \int _{1}^{\infty }f(x)\,dx\,}$

converge together or diverge together.

Notice that if ${\displaystyle f}$ were negative, continuous, and increasing this is also true since such a function would simply be the negative of some function which is positive, continuous, and decreasing and multiplying by -1 will not change the convergence of a series.

solution Discuss the convergence of ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n+1}}}$.

solution Discuss the convergence of ${\displaystyle \sum _{n=0}^{\infty }e^{-3n}}$.

solution Discuss the convergence of ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{5}}}}$.

solution Discuss the convergence of ${\displaystyle \sum _{n=1}^{\infty }{\frac {n}{e^{n}}}}$.

solution Explain why the integral test is or is not applicable to ${\displaystyle \sum _{n=1}^{\infty }n^{2}}$.

solution Explain why the integral test is or is not applicable to ${\displaystyle \sum _{n=1}^{\infty }-{\frac {1}{n^{2}}}}$.

solution Explain why the integral test is or is not applicable to ${\displaystyle \sum _{n=1}^{\infty }{\frac {|\sin n|+1}{\ln(n+1)}}}$.

solution Discuss the convergence of ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{p}}}}$.

solution Discuss the convergence of ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{\frac {7}{3}}}}}$.

solution Discuss the convergence of ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{\sqrt[{3}]{n}}}}$.

### Comparison of Series

 Direct Comparison If ${\displaystyle 0 for all ${\displaystyle n}$ If ${\displaystyle \sum _{n=1}^{\infty }b_{n}}$ converges, then ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ also converges. 2. If ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ diverges, then ${\displaystyle \sum _{n=1}^{\infty }b_{n}}$ also diverges.
 Limit Comparison Suppose ${\displaystyle a_{n}>0}$, ${\displaystyle b_{n}>0}$ and ${\displaystyle \lim _{n\rightarrow \infty }{\frac {a_{n}}{b_{n}}}=L}$ where ${\displaystyle L}$ is finite and positive. Then ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ and ${\displaystyle \sum _{n=1}^{\infty }b_{n}}$ either both converge or both diverge.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{3}+4}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {\ln n}{n-3}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {2n^{2}}{4n^{4}+8n^{3}+3}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n{\sqrt {n^{2}+1}}}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {n+4}{(n+2)(n+1)}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }\sin {\frac {1}{n}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {n+1}{n\times 3^{n+1}}}}$.

### Dirichlet's Test

Let ${\displaystyle a_{k},b_{k}\in \mathbb {R} }$ for ${\displaystyle k\in \mathbb {N} }$.

If the sequence of partial sums ${\displaystyle s_{n}=\sum _{k=1}^{n}a_{k}}$ is bounded and ${\displaystyle b_{k}\downarrow 0}$ as ${\displaystyle k\rightarrow \infty }$, then ${\displaystyle \sum _{k=1}^{\infty }a_{k}b_{k}}$ converges.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=1}^{\infty }{\frac {\sin {\frac {k\pi }{2}}}{k}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=2}^{\infty }{\frac {\sin {\frac {k\pi }{3}}}{\ln k}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=7}^{\infty }\left(-{\frac {1}{2}}\right)^{k}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k2^{k}}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=3}^{\infty }{\frac {1}{\ln(\ln k)}}\cos \left({\frac {k\pi }{3}}\right)}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=1}^{\infty }a_{k}{\frac {1}{k^{3}}}}$ where ${\displaystyle a_{k}=\{7,4,6,3,-10,-10,7,4,6,3,-10,-10,...\}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=1}^{\infty }\sin {\frac {2k\pi }{n}}{\frac {1}{\ln(\ln(\ln k))}}}$ for any integer ${\displaystyle n\geq 1}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{k=3}^{\infty }{\frac {\ln(\ln k)}{\ln k}}a_{k}}$ where ${\displaystyle \{a_{k}\}=\{1,-1,1,2,-3,1,2,4,-7,1,-1,1,2,-3,1,2,4,-7,...\}}$.

### Alternating Series

If ${\displaystyle a_{n}}$, then the alternating series

${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}a_{n}\,}$ and ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}a_{n}\,}$

converge if the absolute value of the terms decreases and goes to ${\displaystyle 0}$.

solution Discuss the convergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n}}\,}$.

solution Discuss the convergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}n^{2}}{n^{2}+1}}\,}$.

solution Discuss the convergence of the series ${\displaystyle \sum _{n=1}^{\infty }\cos(n\pi )\,}$.

solution Discuss the convergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{\sqrt {n}}}\,}$.

solution Discuss the convergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}n^{2}}{e^{n}}}\,}$.

solution Discuss the convergence of ${\displaystyle \sum _{n=2}^{\infty }{\frac {(-1)^{n}}{\ln n}}}$ including whether the sum converges absolutely or conditionally.

solution Discuss the convergence of ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}+3n+4}}}$ including whether the sum converges absolutely or conditionally.

solution Discuss the convergence of ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n!}}}$ including whether the sum converges absolutely or conditionally.

### Ratio Test

For the infinite series ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$,

1. If ${\displaystyle 0\leq \lim _{n\rightarrow \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|<1\,}$, then the series converges absolutely.

2. If ${\displaystyle \lim _{n\rightarrow \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|>1\,}$, then the series diverges.

3. If ${\displaystyle \lim _{n\rightarrow \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|=1\,}$, then the test is inconclusive.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {2^{n}}{(2n)!}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {n}{3^{n+1}}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{3}}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {3^{n}}{n^{2}+2}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {n}{(n+2)!}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {n4^{n}}{n!}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {n!}{4^{n}}}}$.

solution Discuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {n^{k}a^{n}}{n!}}}$.

### Root Test

Let ${\displaystyle a_{k}\in \mathbb {R} }$ and ${\displaystyle r=\limsup _{k\rightarrow \infty }|a_{k}|^{\frac {1}{k}}}$

1. If ${\displaystyle r<1\,}$, then ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ converges absolutely.

2. If ${\displaystyle r>1\,}$, then ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ diverges.

3. If ${\displaystyle r=1\,}$, this test is inconclusive.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }\left({\frac {n}{2n+1}}\right)^{n}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }e^{-n}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {n}{3^{n}}}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }\left({\frac {2n}{100}}\right)^{n}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {(n!)^{n}}{(n^{n})^{2}}}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }\left({\frac {7n}{12n-6}}\right)^{2n}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }\left({\frac {15n-6}{4n+2}}\right)^{7n}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {n!}{(n+1)^{n}}}\left({\frac {23}{50}}\right)^{n}}$.

### Cauchy Condensation Test

The series ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ and ${\displaystyle \sum _{k=1}^{\infty }2^{k}a_{2^{k}}}$ converge or diverge together.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k}}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{\ln k}}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{\sqrt {2^{k}}}}}$.

solutionDiscuss the convergence or divergence of the series ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{(2^{k})^{n}}}}$, where n is some real number.

## Series of Real Functions

solution Find the infinite series expansion of ${\displaystyle {\frac {1}{(1+x)^{a}}}\,}$

solution Investigate the convergence of this series: ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k(k+1)}}\,}$

solution Investigate the convergence of this series: ${\displaystyle 3-2+{\frac {4}{3}}-{\frac {8}{9}}+...+3\left(-{\frac {2}{3}}\right)^{k}+...\,}$

solution Find the upper limit of the sequence ${\displaystyle \left\{x_{n}\right\}_{n=1}^{\infty },x_{n}=(-1)^{n}n\,}$

solution Find the upper limit of the sequence ${\displaystyle \left\{x_{n}\right\}_{n=1}^{\infty },x_{n}=(-1)^{n}\left({\frac {2n}{n+1}}\right)\,}$

solution Evaluate ${\displaystyle \sum _{k=0}^{n}{n \choose k}^{2}\,}$

solution Evaluate ${\displaystyle \sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {m^{2}\,n}{3^{m}\left(m\,3^{n}+n\,3^{m}\right)}}}$.

solution Find the upper limit of the sequence ${\displaystyle \left\{x_{n}\right\}_{n=1}^{\infty },x_{n}={\frac {1}{n^{2}}}\,}$

solution Find the upper limit of the sequence ${\displaystyle \left\{x_{n}\right\}_{n=1}^{\infty },x_{n}=n\sin \left({\frac {n\pi }{2}}\right)\,}$

solution Evaluate ${\displaystyle \sum _{n=0}^{\infty }\left({\frac {i}{3}}\right)^{n}\,}$

solution Determine the interval of convergence for the power series ${\displaystyle \sum _{n=0}^{\infty }{\frac {n(3x-4)^{n}}{{\sqrt[{3}]{n^{4}}}(2x)^{n-1}}}.}$