Series convergence

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?

Dick
Homework Helper

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.

actaully i don't want to find the limit of the fn. i want the sum of the series.

Dick
Homework Helper
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:
Ray Vickson
Homework Helper
Dearly Missed

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"
$$\left|A_n + \frac{1}{A_n} \right|$$
(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
$$\sum_{k=0}^n \frac{k}{n^2+k} = n+1+n^2 \Psi(n^2)-n^2 \Psi(n^2+n+1)$$
where ##\Psi(x)## is the so-called "Psi function" or "di-Gamma function", defined as the logarithmic derivative of the Gamma function ##\Gamma(x)##:
$$\Psi(x) \equiv \frac{d \ln(\Gamma(x))}{dx} = \frac{\Gamma\, ^{\prime} (x)}{\Gamma(x)}.$$

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@ray wickson,
could you tell me an answer for this.

Dick