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MCarsten

Member

From: Brazil

Registered: 2017-10-01

Posts: 1

Sequence convergence and limit

The problem read as follows:

Determine if the sequence

converges or diverges. If it converges, give its limit value.

The answer is: converges, the limit is zero.

I know that this sequence limit can be evaluated easily with some algebra manipulation getting the corresponding function and applying L'Hopital's rule to it. But I would like to try another way around just for the sake of playing with something less mechanical.

Below is my reasoning. I'd be grateful if someone can point me out if it is correct.

---

I have tried the following:

First, I've evaluated a few terms looking forward to eventually fathom some pattern. So:

As you can see, each new term equals the next index to the alpha-th power subtracted from the sum of all

terms plus one.

Subtracting

from both sides yields:

Let

be the sum of

terms. We have:

If we take the roots on both sides we get:

It is given by the problem that

, therefore

.

Substitute

in our

:

Since we are dealing with a limit process, we're interested when $m$ is a very large number, that is, when $m \gg 1$. In this particular, the second $+1$ in the first term above renders no effect. Therefore we can discard him. We will get:

. We can conclude that the limit is $0$, that is:

and the given sequence converges.

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,440

Website

Re: Sequence convergence and limit

Hi MCarsten,

Welcome to the forum.

Your reasoning is sound up until the last paragraph:

Since we are dealing with a limit process, we're interested when $m$ is a very large number, that is, when $m \gg 1$. In this particular, the second $+1$ in the first term above renders no effect. Therefore we can discard him.

Isn't this the point of the question? By your argument, you could apply the same reasoning to

, no?