Series converge/ diverges. determine sum of series

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Discussion Overview

The discussion revolves around determining the convergence or divergence of the series defined by the summation from n=1 to infinity of 2/n(n+2), and if it converges, finding its sum analytically. The scope includes mathematical reasoning and homework-related inquiry.

Discussion Character

  • Homework-related, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant proposes using partial fractions to analyze the series, suggesting that it can be expressed as a telescoping series.
  • Another participant points out that the original work contains an error in the partial fraction decomposition, indicating that the correct form involves subtraction rather than addition.
  • A later reply suggests that despite the error, the initial participant arrived at the correct sum of the series.
  • One participant later claims to have recalculated the sum as 3/2, indicating a potential correction or alternative conclusion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct form of the partial fraction decomposition, and there is disagreement regarding the final sum of the series, with different values proposed.

Contextual Notes

There are unresolved issues regarding the accuracy of the mathematical steps taken, particularly in the partial fraction decomposition and the implications for the series sum.

mattmannmf
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Determine if the series converges or diverges ad if it converges then determine the sum of the series analytically:

infinity
{Sigma} 2/n(n+2)
n=1

so i used partial fractions and got:
{Sigma} [1/n + 1/(n+2)]

then i used telescoping form to get the nth partial sum...
Sn= (1/1 + 1/3) + (1/2 + 1/4) +...

then i got the nth partial sum to be = 1+1/(n+2)

so the series converges and its sum is 1?

Does that seem right to everyone?
 
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First, this is in the wrong section, I believe. It should go in the homework and coursework area.

Second, yes. However, be careful...

As you did the work wrong, and yet got the right answer. The partial fraction decomposition for \frac{2}{(n)(n+2)} isn't quite what you posted. Can you see the error?
 
oh ok. i thought this was the homework and course area.

yea its supposed to be subtraction, not addition. i got 3/2 to be the sum of the series
 
There you go.

You win...
 

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