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## Homework Statement

Suppose we are given the power series expansion ##f(x) = -\sum_{n=1}^{\infty} \frac{x^{n}}{n} ## which converges for |z|<1.

What is the radius of convergence?

Sum this serie and derive a power series expansion for the resulting function around -1/2, 1/2, 3/4 and 2.

## The Attempt at a Solution

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I can let ##f(x) = \sum_{n=0}^{\infty} \frac{x^{n}+1}{n+1}## so ##f'(x) = x^n = \frac{1}{1-x}## Integrating back gives ##f(x) = -ln(\frac{1}{1-x})=-\sum_{n=1}^{\infty} \frac{x^n}{n} ##

Now I can just plug in the values -1/2, 1/2, 3/4 and 2 right?

My question is, why do they ask me to do this with many values if all I need to do is plug it in in the final equation? I think I probably need to do something else/more...

Also, how do I get the radius of convergence? I recall it is 1 in this case, but not how? I've heard I can take the inverse of the results of the ratio test, but that's gives me an expression, ##\frac{x_n}{n+1}## and not a number?