Calculus 2 - Infinite Series

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Determine weather or not the following series converges or diverges.

sum[k=1,inf] 2/(k^2-1)

I applied the limit comparison test with 1/k^2

2* lim k->inf [ 1/(k^2-1) ] /(1/k^2) = 2*lim k->inf k^2/(k^2-1) =H 2

then because

sum[k=1,inf] 1/k^2 is a convergent p-series then sum[k=1,inf] 2/(k^2-1) must also converge by the limit comparison test.

I plugged this into wolfram alpha and it said that the sum does not converge. Am I doing something wrong? Thanks for any help.

This is what I put into wolfram alpha
sum[n=1,inf] of 2/(k^2-1)
 

Answers and Replies

  • #2
Dick
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Determine weather or not the following series converges or diverges.

sum[k=1,inf] 2/(k^2-1)

I applied the limit comparison test with 1/k^2

2* lim k->inf [ 1/(k^2-1) ] /(1/k^2) = 2*lim k->inf k^2/(k^2-1) =H 2

then because

sum[k=1,inf] 1/k^2 is a convergent p-series then sum[k=1,inf] 2/(k^2-1) must also converge by the limit comparison test.

I plugged this into wolfram alpha and it said that the sum does not converge. Am I doing something wrong? Thanks for any help.

This is what I put into wolfram alpha
sum[n=1,inf] of 2/(k^2-1)
Oh, I see what you are doing. Do you really mean [n=1,inf]?? n?? And the k=1 term of your series is undefined. Skip it.
 
  • #3
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so I changed it to
sum[k=1,inf] of 2/(k^2-1)
and it says the sum does not converge
 
  • #4
Dick
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so I changed it to
sum[k=1,inf] of 2/(k^2-1)
and it says the sum does not converge
I told you. The k=1 term is undefined. That's probably WA's problem with that one.
 
  • #5
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I told you. The k=1 term is undefined. That's probably WA's problem with that one.
Oh. Thanks.
 
  • #6
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I told you. The k=1 term is undefined. That's probably WA's problem with that one.
Might be a case of GIGO, or "garbage in, garbage out."
 

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