Series converge/ diverges. determine sum of series

[mattmannmf]

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?

 

[Char. Limit]

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?

 

[mattmannmf]

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

 

[Char. Limit]

There you go.

You win...

 

[CellCoree]

I would say that your approach seems sound and your conclusion is correct. However, it is always important to check your work and make sure that your solution is mathematically valid. In this case, you could also use the Ratio Test or the Comparison Test to confirm that the series converges. Additionally, you could also use a computer program or calculator to calculate the sum of the series and compare it to your analytical solution. Overall, your method is a valid way to determine the convergence and sum of the given series.