Some help with finding the limit of a series

[vande060]

Homework Statement

determine whether or not the sequence converges of diverges. If it converges, find the limit.

a_n = ((-1)^n-1n)/(n² + 1)

Homework Equations

I have a feeling this theorem is used

if lim _n->∞ |a_n| = 0, then lim _n->∞ a_n = 0

The Attempt at a Solution

I don't really know where to start on this one, I thought about using L'Hospital for this, but I don't know how to do exponential derivatives without taking a log, and I don't remember the teacher ever doing that in class when looking at this variety of problem

 

[Mark44]

Homework Statement

determine whether or not the sequence converges of diverges. If it converges, find the limit.

a_n = ((-1)^n-1n)/(n² + 1)

Homework Equations

I have a feeling this theorem is used

if lim _n->∞ |a_n| = 0, then lim _n->∞ a_n = 0

The Attempt at a Solution

I don't really know where to start on this one, I thought about using L'Hospital for this, but I don't know how to do exponential derivatives without taking a log, and I don't remember the teacher ever doing that in class when looking at this variety of problem

Do you have any sense of what this one does?
b_n = n/(n² + 1)

For n > 0, the denominator is always larger than the numerator, and gets larger more rapidly.

 

[vande060]

Do you have any sense of what this one does?
b_n = n/(n² + 1)

For n > 0, the denominator is always larger than the numerator, and gets larger more rapidly.

yes i understand that, that would mean that the series is decreasing right?

 

[Mark44]

You seem to be using the terms "series" and "sequence" interchangeable, but they are different. Your thread title includes the word series, but what you showed is a sequence.

For your sequence, it might help to pull a factor of n² out of the numerator, like so:
$$a_n = (-1)^{n - 1}\frac{n}{n^2(1 + 1/n^2)}$$

This really isn't a very complicated sequence.

 

[vande060]

You seem to be using the terms "series" and "sequence" interchangeable, but they are different. Your thread title includes the word series, but what you showed is a sequence.

For your sequence, it might help to pull a factor of n² out of the numerator, like so:
$$a_n = (-1)^{n - 1}\frac{n}{n^2(1 + 1/n^2)}$$

This really isn't a very complicated sequence.

Im sorry, I am sure its not complicated, but this is one of the first ones I've seen

I think I understand how to do the rest now though.

If I find the limit of the absolute value of this sequence I find that it is zero. By the theorem I posted earlier, the limit of the original sequence with the -1 is also 0. Correct?

Thanks for your help so far

 

[Mark44]

Yes, the limit of the absolute values of this sequence is zero, which means that the limit of the original sequence is also zero.