Infinite sum converge to what value?

[pivoxa15]

Homework Statement

The infinite series (-1)^n(x/n) from n=1 converges. But what is the specific value of it?

 

[Gib Z]

\ln(1+x) = \sum^{\infty}_{n=0} \frac{(-1)^n}{n+1} x^{n+1}
\sum_{n=1}^{\infty} (-1)^n\frac{x}{n} = x\sum_{n=1}^{\infty} \frac{(-1)^n}{n}=x\log_e 2

 

[ILEW]

\ln(1+x) = \sum^{\infty}_{n=0} \frac{(-1)^n}{n+1} x^{n+1}
\sum_{n=1}^{\infty} (-1)^n\frac{x}{n} = x\sum_{n=1}^{\infty} \frac{(-1)^n}{n}=x\log_e 2

You have put x=1 so it should just be ln(2)?

 

[Gib Z]

Yup. Exactly.

 

[pivoxa15]

But ln(2)>0 and \sum_{n=1}^{\infty} (-1)^n\frac{x}{n}<0 since the first term is negative and has the largest magnitude so will dominate the series. The series should equal -ln(2) so you may have made an error with your series manipulation.

 

[Gib Z]

Yea sorry about that >.< I made a mistake with the starts of the series, some were n=1 and others n=0, and I didn't handle them well. But youve got the idea