Silenthill
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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Silenthill
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\[\sum_{n=1}^{\infty} ((1)^{n+1} x^n)/n \]

blockcolder
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That's ln(1+x).

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

blockcolder
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Alternating series test.

joemath314159
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Are you familiar with Taylor Series or Maclaurin Series?

Silenthill
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yes

joemath314159
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If you try to compute the Maclaurin series of:\[\ln(x+1)\]you will obtain the power series in your question.

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

Zarkon
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*integrate the closed form after summing the geometric