## anonymous 4 years ago Find the sum of the series and explain how you know the sum converges to this value. Sum of (((-1)^(n+1))/n) n=1, infinty.

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1. anonymous

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

2. blockcolder

That's ln(1+x).

3. anonymous

ln(1+1)=ln(2) but how you know it converges to this value?

4. blockcolder

Alternating series test.

5. anonymous

Are you familiar with Taylor Series or Maclaurin Series?

6. anonymous

yes

7. anonymous

If you try to compute the Maclaurin series of:$\ln(x+1)$you will obtain the power series in your question.

8. Zarkon

differentiate the series...sun the then geometric series...integrate the series

9. Zarkon

*integrate the closed form after summing the geometric