Infinite sum converge to what value?

1. Apr 9, 2007

pivoxa15

1. The problem statement, all variables and given/known data
The infinite series (-1)^n(x/n) from n=1 converges. But what is the specific value of it?

2. Apr 9, 2007

Gib Z

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

3. Apr 9, 2007

ILEW

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

4. Apr 9, 2007

Gib Z

Yup. Exactly.

5. Apr 9, 2007

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.

Last edited: Apr 9, 2007
6. Apr 10, 2007

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

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