Calculating the sum $\sum \frac{x^n}{n}$.

Question

I am asked to calculate the inverval of convergence and calculate the sum of $$\sum \frac{x^n}{n}$$

I've seen that it converges for $-1<x<1$, but how do I calculate the sum? For which $x$?

See https://math.stackexchange.com/questions/1356517/what-is-the-correct-radius-of-convergence-for-ln1x — lab bhattacharjee, Nov 07 '17 at 14:22

score 2 · Accepted Answer · answered Nov 07 '17 at 13:59

2

Note that for $|x|<1$

$$\sum_{n=1}^{\infty} \frac{x^n}{n} = \int \sum x^{n-1}dx= \int \frac{1}{1-x} dx = -\log(1-x)+c$$

Now it is easy to determine $c$ by plugging in e.g. $x=0$.

answered Nov 07 '17 at 13:59

flawr

I understand all the equalities, but how does plugging any $|x|<1$ help me find $c$? – John Keeper Nov 07 '17 at 14:10
You will get an equation that just involves $c$ as an unknown, which you can solve for. – flawr Nov 07 '17 at 14:11

1 Answers1