Calc6.6

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

The integral test states a function must be continuous, positive, and decreasing. This function is continuous but it is negative and increasing. However, we can easily use the integral test. Instead, look at the function f(x)={\frac  {1}{x^{2}}}. This function is continuous, positive, and decreasing on the interval [1,\infty ] so the integral test applies. In fact, the integral test confirms that its series converges. Thus, since this series converges, multiplying every term by -1 will simply change the sign of the sum but will not affect convergence at all and this is all we need to show to know that the given series converges.


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