Solving the Limit of a[n] as n Goes to Infinity

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Homework Help Overview

The discussion revolves around finding the limit of a sequence as n approaches infinity, specifically focusing on the convergence of a series. The original poster (OP) attempts to analyze the sequence using Riemann sums and integrals but expresses difficulty in determining a starting point for the limit calculation.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the nature of the series, questioning whether it can be classified as a geometric series or a telescoping sum. Some suggest using the ratio test and separating functions to facilitate the summation process. Others propose differentiating a geometric series as a potential method for finding the limit.

Discussion Status

The conversation is ongoing, with several participants providing hints and suggestions for approaches without reaching a consensus on a specific method. The OP has indicated progress in understanding the differentiation of geometric series, but a complete solution has not yet been presented.

Contextual Notes

There is a recurring emphasis on the conditions under which the series converges, with participants questioning the assumptions related to the convergence criteria and the nature of the series itself.

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Homework Statement


[PLAIN]http://admitere.ncit.pub.ro/moodle/filter/tex/pix.php/5c2cef253f2db3240db03f8c9b6c9463.gif

lim n->infinity of a[n] = ?

Homework Equations


|x| > 1

The Attempt at a Solution


Well, actually i figured out that the sequence converges, and I've tried to solve it using riemannian sums and integrals but no success.

I've also tried to solve the sum and then calculate the limit but i had no idea where to start from.

May anyone suggest me a starting point?
 
Last edited by a moderator:
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I think we can find the sum of a series only if it is a geometric series or a telescoping sum, and from a quick glance this doesn't look like either one.
Otherwise we would need some math software to approximate the sum.
 
flyerpower said:

Homework Statement


[PLAIN]http://admitere.ncit.pub.ro/moodle/filter/tex/pix.php/5c2cef253f2db3240db03f8c9b6c9463.gif

lim n->infinity of a[n] = ?

Homework Equations


|x| > 1


The Attempt at a Solution


Well, actually i figured out that the sequence converges, and I've tried to solve it using riemannian sums and integrals but no success.

I've also tried to solve the sum and then calculate the limit but i had no idea where to start from.

May anyone suggest me a starting point?

Hint: think about what happens when you take derivatives of a geometric series.
 
Last edited by a moderator:
flyerpower said:

Homework Statement


[PLAIN]http://admitere.ncit.pub.ro/moodle/filter/tex/pix.php/5c2cef253f2db3240db03f8c9b6c9463.gif

lim n->infinity of a[n] = ?

Homework Equations


|x| > 1

The Attempt at a Solution


Well, actually i figured out that the sequence converges, and I've tried to solve it using riemannian sums and integrals but no success.

I've also tried to solve the sum and then calculate the limit but i had no idea where to start from.

May anyone suggest me a starting point?

use the ratio test. ignore the summation symbol for now.

After using ratio test. see if you can separate the functions so you have "(function 1) + or - (function 2)"

if you can do that, then you could probably telescope to solve the sum. If not, then it could be a geometric series, which should be easier to test (if it is or not) after u get it to a simplified form.

If that doesn't work you should retry it but changing the index. i.e do an index shift, sub k=K+1 (them summation will start at 0), index will be K, not k-1. ratio test should be easier

(also, isn't this question supposed to be something like, for what values of x or k, is this covnergant?)
 
Last edited by a moderator:
cloud360 said:
use the ratio test. ignore the summation symbol for now.

After using ratio test. see if you can separate the functions so you have "(function 1) + or - (function 2)"

if you can do that, then you could probably telescope to solve the sum. If not, then it could be a geometric series, which should be easier to test (if it is or not) after u get it to a simplified form.

If that doesn't work you should retry it but changing the index. i.e do an index shift, sub k=K+1 (them summation will start at 0), index will be K, not k-1. ratio test should be easier

(also, isn't this question supposed to be something like, for what values of x or k, is this covnergant?)
The OP has already determined that the series converges. He/she is now trying to figure out what the series converges to.
 
My hint still stands. Figure out the sum of r^(k+1). Differentiate twice and put r=1/x.
 
I figured out how to do it differntiating a geometric series after Dick gave me that hint but i hadn't time to do it yesterday, i will come back later with the solution.
 
magicarpet512 said:
I think we can find the sum of a series only if it is a geometric series or a telescoping sum

No, that's not true. For example, the sum of 1/n! is e.
 

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