# Divergence of a particular serie

• ShizukaSm
In summary, the series \sum_{n=1}^{\infty}{(-1)^n*n} diverges by the Test for Divergence because the limit does not exist, and thus the series diverges.
ShizukaSm

## Homework Statement

$\sum_{n=1}^{\infty}{(-1)^n*n}$

## The Attempt at a Solution

Well, this is a series I came upon when analyzing the endpoints of a particular power series, the thing is, my book says it's divergent by the Test for divergence, however, I can't find this result, what I tried to do so far:

Alternating testing series: Of course, my first thought, the test fails because $b_{n+1} > b_n$, and not the opposite.

Then I tried the test for divergence:
$\lim_{n->\infty}{(-1)^n*n}$
But I can't solve this limit, I could solve it without the $(-1)^n$. But I can't just "take out the term I don't like and solve it".

What am I doing wrong?

ShizukaSm said:

## Homework Statement

$\sum_{n=1}^{\infty}{(-1)^n*n}$

## The Attempt at a Solution

Well, this is a series I came upon when analyzing the endpoints of a particular power series, the thing is, my book says it's divergent by the Test for divergence, however, I can't find this result
Some books call this the "Nth term test for divergence." This theorem is very simple to use, and should probably be used first. It says that in a series ##\sum a_n##, if ##\lim_{n \to \infty} a_n ≠ 0##, then the series diverges.

Note that this theorem can be used only on series for which ##\lim_{n \to \infty} a_n ≠ 0##. If the limit equals zero, the series could converge or it could diverge.
ShizukaSm said:
, what I tried to do so far:

Alternating testing series: Of course, my first thought, the test fails because $b_{n+1} > b_n$, and not the opposite.
This test doesn't fail - it is not applicable here.
ShizukaSm said:
Then I tried the test for divergence:
$\lim_{n->\infty}{(-1)^n*n}$
But I can't solve this limit, I could solve it without the $(-1)^n$. But I can't just "take out the term I don't like and solve it".
Expand the series by writing a few of the terms. It should be obvious what this series is doing.
ShizukaSm said:
What am I doing wrong?

Last edited:
Yes indeed, I can see that it obviously diverge if I expand the series, I even tried graphing a few terms to look at that, but what I meant was, is there a "formal way" to write that?

ArcanaNoir said:
I think the divergence test says "if the limit is not 0 or if the limit does not exist, then the series diverges."
If the limit does not exist, it is automatically not zero, so it isn't necessary to include the part about it not existing. Perhaps it is added above for clarity.
ArcanaNoir said:
Here's a website to back that up: http://www.mathscoop.com/calculus/infinite-sequences-and-series/nth-term-test.php
I'd check a calculus book but I don't have one with me.

ShizukaSm said:
Yes indeed, I can see that it obviously diverge if I expand the series, I even tried graphing a few terms to look at that, but what I meant was, is there a "formal way" to write that?

You would write something like "since the limit doesn't exist, the series diverges by the divergence test".

## 1. What is the definition of divergence of a series?

The divergence of a series is a measure of its tendency to approach infinity or negative infinity as the number of terms increases. It is determined by evaluating the limit of the partial sums of the series.

## 2. How do you determine if a series diverges?

A series diverges if the limit of its partial sums is infinite or undefined. This can be determined by using various convergence tests, such as the comparison test or the ratio test.

## 3. Can a series diverge to a specific value?

No, a series can only diverge to infinity or negative infinity. If the limit of the partial sums is a finite number, then the series is said to converge to that value.

## 4. What is the significance of divergence of a series in mathematics?

The divergence of a series is important in determining the behavior of the series, as well as its convergence or divergence. It also plays a role in various mathematical concepts, such as infinite series and improper integrals.

## 5. How does the divergence of a series relate to the terms of the series?

The divergence of a series is determined by the behavior of its terms. If the terms of a series do not approach zero as the number of terms increases, then the series will diverge. However, it is possible for a series with terms that approach zero to still diverge if the terms do not decrease fast enough.

Replies
3
Views
542
Replies
6
Views
754
Replies
2
Views
680
Replies
2
Views
1K
Replies
5
Views
1K
Replies
2
Views
1K
Replies
16
Views
1K
Replies
9
Views
1K
Replies
1
Views
908
Replies
7
Views
2K