Alg9.3.3

From Exampleproblems

Find the sum of the infinite series $\frac{1}{2}-\frac{1}{2\cdot2^2}+\frac{1}{3\cdot2^3}-.........\,$

Readjusting the values $\frac{1}{2}-\frac{1}{2\cdot2^2}+\frac{1}{3\cdot2^3}-.........= \frac{1}{2}-\frac{[\frac{1}{2}]^2}{2}+\frac{[\frac{1}{2}]^3}{3}-\frac{[\frac{1}{2}]^4}{4}+........\,$

$\frac{1}{2}-\frac{1}{2\cdot2^2}+\frac{1}{3\cdot2^3}-.........= \mathrm{Log}(1+\frac{1}{2})\,$

Further simplified to $\frac{1}{2}-\frac{1}{2\cdot2^2}+\frac{1}{3\cdot2^3}-.........= \mathrm{Log}_e(\frac{3}{2})\,$

$\frac{1}{2}-\frac{1}{2\cdot2^2}+\frac{1}{3\cdot2^3}-.........= \mathrm{Log}_e(1\cdot5) \,$