Question

Find the sum of the following series.

Answer 1

First, write the series as a power series. It's geometric with \(x=-\dfrac{1}{2}\), so
\[-\sum_{n=1}^\infty \frac{1}{n}\left(-\frac{1}{2}\right)^n~~\implies~~f(x)=-\sum_{n=1}^\infty \frac{x^n}{n}\]
Try taking the derivative.

Answer 2

- series of x^(n-1)

Answer 3

Right,
\[f'(x)=-\sum_{n=1}^\infty x^{n-1}=-\sum_{n=0}^\infty x^n\]
For \(|x|<1\), what's the value of this series?

Answer 4

- 1/ (1-x)*

Answer 5

Yup, so now you can take the antiderivative to find the function represented by the power series.

Answer 6

ln(1-x)

Answer 7

Alright, so you know that \(f(x)=\ln(1-x)\). The given series gives the value for \(x=-\dfrac{1}{2}\), so what's \(f\left(-\dfrac{1}{2}\right)\)?

Answer 8

I see! awesome thank you

Answer 9

You're welcome!