anonymous
  • anonymous
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.
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
\[\sum_{n=1}^{\infty} ((-1)^{n+1} x^n)/n \]
blockcolder
  • blockcolder
That's ln(1+x).
anonymous
  • anonymous
ln(1+1)=ln(2) but how you know it converges to this value?

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blockcolder
  • blockcolder
Alternating series test.
anonymous
  • anonymous
Are you familiar with Taylor Series or Maclaurin Series?
anonymous
  • anonymous
yes
anonymous
  • anonymous
If you try to compute the Maclaurin series of:\[\ln(x+1)\]you will obtain the power series in your question.
Zarkon
  • Zarkon
differentiate the series...sun the then geometric series...integrate the series
Zarkon
  • Zarkon
*integrate the closed form after summing the geometric

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