Finding A Function From A Power Series

[Bashyboy]

Homework Statement

Find the sum of the series and its radius of convergence:

\sum_{n=1}^{\infty} (-1)^{n+1}\frac{(x-1)^n}{n}

Homework Equations

The Attempt at a Solution

I found the radius of convergence, but I wasn't sure how to find the sum of the power series.

 

[LCKurtz]

Homework Statement

Find the sum of the series and its radius of convergence:

\sum_{n=1}^{\infty} (-1)^{n+1}\frac{(x-1)^n}{n}

Homework Equations

The Attempt at a Solution

I found the radius of convergence, but I wasn't sure how to find the sum of the power series.

Hint: If you call that series f(x), what is f'(x)?

 

[Bashyboy]

Would it be f'(x) = \sum_{n=0}^{\infty} (-1)^{n+1}(x-1)^{n-1}?

 

[LCKurtz]

Would it be f'(x) = \sum_{n=0}^{\infty} (-1)^{n+1}(x-1)^{n-1}?

Yes. And do you recognize what kind of series that is?

 

[Bashyboy]

If you distribute the (-1)^(n+1) to the (x-1)^(n-1) would it be a geometric series?

 

[LCKurtz]

Would it be f'(x) = \sum_{n=0}^{\infty} (-1)^{n+1}(x-1)^{n-1}?

If you distribute the (-1)^(n+1) to the (x-1)^(n-1) would it be a geometric series?

Do you really have to ask? What do you think? Why? Show us what you would do if it is.