series..diverging, converging

**champrock** · March 3rd 2009, 08:46 AM

Q2004.
Can you help me with this question..

**TheEmptySet** · March 3rd 2009, 10:15 AM

Originally Posted by champrock

Q2004.
Can you help me with this question..

Note that $\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$

so this telescopes to 1. So it is none of these.

**mateoc15** · March 3rd 2009, 10:25 AM

This seems like a good candidate for "plug and chug". Just try a few values and see what happens. You'll figure it out.

**champrock** · March 3rd 2009, 05:36 PM

Originally Posted by TheEmptySet

Note that $\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$

so this telescopes to 1. So it is none of these.

i has used your method and found the value to be 1-Lim(1/n+1) where n tends to infinity.

So this becomes 1-(some minutest number close to 0) . so shouldnt the answer be "some number between 0 and 1" ?

i am confused on this part.

**TheEmptySet** · March 4th 2009, 03:58 PM

Originally Posted by champrock

i has used your method and found the value to be 1-Lim(1/n+1) where n tends to infinity.

So this becomes 1-(some minutest number close to 0) . so shouldnt the answer be "some number between 0 and 1" ?

i am confused on this part.

$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=\sum_{n=1}^{\i nfty}\frac{1}{n}-\frac{1}{n+1}=\sum_{n=1}^{\infty}\frac{1}{n}-\sum_{n=1}^{\infty}\frac{1}{n+1}=$

Now lets take out the first term of the series

$1+\sum_{n=2}^{\infty}\frac{1}{n}-\sum_{n=1}^{\infty}\frac{1}{n+1}$

Now lets reindex the first sum with $k=n-1 \iff n=k+1$

Note: dont forget that the index is a "dummy variable"

$1+\sum_{k=1}^{\infty}\frac{1}{k+1}-\sum_{n=1}^{\infty}\frac{1}{n+1}$

Since the index is a dummy vairable we can combine the sums let k=n

$1+\sum_{n=1}^{\infty}\frac{1}{n+1}-\sum_{n=1}^{\infty}\frac{1}{n+1}=1+\sum_{n=1}^{\in fty}\left(\frac{1}{n+1}-\frac{1}{n-1} \right)=1+\sum_{n=1}^{\infty}0=1$

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