Series convergence

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  • #1
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Homework Statement



An=Ʃ(k)/[(n^2)+k]
the sum is k=0 to n, the question is, to which value does the this series converge to

Homework Equations


i know for sure that this series converges, but could not figure out the value to whch it converges


The Attempt at a Solution



i did the convergence test, mod(An+1/An).. the value is 1. what i do now?
 

Answers and Replies

  • #2
Dick
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Homework Statement



An=Ʃ(k)/[(n^2)+k]
the sum is k=0 to n, the question is, to which value does the this series converge to

Homework Equations


i know for sure that this series converges, but could not figure out the value to whch it converges


The Attempt at a Solution



i did the convergence test, mod(An+1/An).. the value is 1. what i do now?

You want to use the squeeze theorem. Try and bracket that sum between two sums whose limit you can evaluate and whose limit turn out to be the same. Here's a hint. The k in the denominator is what's making it hard to evaluate.
 
  • #3
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actaully i don't want to find the limit of the fn. i want the sum of the series.
 
  • #4
Dick
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actaully i don't want to find the limit of the fn. i want the sum of the series.

'converges' means you take the limit of the sums ##A_n## as n->infinity. There's no simple closed form expression for the sum. There is a simple expression for the limit of the sum. This isn't really the same as the usual infinite sum problem. ##A_n## isn't the partial sum of some series. The upper limit n is in the expression for the terms you are summing.
 
Last edited:
  • #5
Ray Vickson
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Homework Statement



An=Ʃ(k)/[(n^2)+k]
the sum is k=0 to n, the question is, to which value does the this series converge to

Homework Equations


i know for sure that this series converges, but could not figure out the value to whch it converges


The Attempt at a Solution



i did the convergence test, mod(An+1/An).. the value is 1. what i do now?

Using the "test"
[tex] \left|A_n + \frac{1}{A_n} \right| [/tex]
(which is what you WROTE) will get you nowhere. Even the correct test ##A_{n+1}/A_n## (written as A_{n+1}/A_n or A_(n+1)/A_n) will still get you nowhere: your problem is NOT to decide on convergence of an infinite series.

The summation cannot be performed in elementary terms: Maple gets the answer
[tex] \sum_{k=0}^n \frac{k}{n^2+k} = n+1+n^2 \Psi(n^2)-n^2 \Psi(n^2+n+1)[/tex]
where ##\Psi(x)## is the so-called "Psi function" or "di-Gamma function", defined as the logarithmic derivative of the Gamma function ##\Gamma(x)##:
[tex] \Psi(x) \equiv \frac{d \ln(\Gamma(x))}{dx} = \frac{\Gamma\, ^{\prime} (x)}{\Gamma(x)}.[/tex]
 
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  • #6
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@ray wickson,
could you tell me an answer for this.
 
  • #7
Dick
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@ray wickson,
could you tell me an answer for this.

Nobody is going to 'tell you an answer', you have to work on it. You really don't need any nonelementary functions for this. Just think about comparison series.
 

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