# Limit of a subsequence

I am trying to calculate limit for a subsequence <${ a }_{ {n }^{ 2} }$> because I was trying to see that limits of sequences and subsequences are both same.

I was able to generate the sum for square values as,

N[ Sum[ 1/(n (n + 1)), {n, Table[n^2, {n, 1, 1000}]}]]


But I want to know how I shall find limit to a subsequence of square integers.

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You can proceed this way (to calculate the sum of the first k terms of the subsequence) :

Sum[ 1/(n (n + 1)) /. n -> n^2, {n, k}]

1/6 (Pi^2 - 3 I PolyGamma[0, 1 - I] + 3 I PolyGamma[0, 1 + I] +
3 I PolyGamma[0, (1 - I) + k] - 3 I PolyGamma[0, (1 + I) + k]
- 6 PolyGamma[1, 1 + k]  )


and take the limit simply :

Limit[ Sum[ 1/(n (n + 1)) /. n -> n^2, {n, k}], k -> Infinity]

 1/6 (3 + Pi^2 - 3 Pi Coth[Pi])


numerically :

N @ %

0.56826

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actually what the final result here is the sum of the subsequence that I got after running the command I have referred to. I need to find Limit of a sequence whose domain is set of n^2. –  Rorschach Apr 6 '13 at 18:13
@rafiki Can't you see this is the way to go ? Perhaps you want simply Limit[1/(n (n + 1)) /. n -> n^2, n -> Infinity]. –  Artes Apr 6 '13 at 18:26
If I am not wrong /. is replacement. But there is difference in calculating a sequence on 1/(n(n+1)) and 1/(n^2(n^2+1))...please correct me if am wrong. –  Rorschach Apr 6 '13 at 18:35
@rafiki Subsequence of $n^2$ terms of $\frac{1}{n(n+1)}$ is $\frac{1}{n^2 (n^2 + 1)}$. What do you mean by calculating of subsequence ? –  Artes Apr 6 '13 at 18:40
subsequence of a sequence belongs to the sequence....it doesn't ! –  Rorschach Apr 6 '13 at 18:43