Proving a Series Conditionally Converges

[I Love Pi]

Hi! I am havping difficulty proving the alternate series converges which is necessary to prove the series is conditionally convergent. Could you please help me spot my error(s)? Thank you very much!

 

[pka]

The alternating series test. Read that link.

 

[I Love Pi]

The link you attached confirms what I applied in my attempt at the problem, however, I am still confused on why the Alternating Series Test failed and how to correct this.

 

[BigBeachBanana]

The link you attached confirms what I applied in my attempt at the problem, however, I am still confused on why the Alternating Series Test failed and how to correct this.

Your [imath]b_n[/imath] should just be [imath]\frac{n}{n^2+1}[/imath] without the minus. To show the sequence is decreasing, consider the function [imath]f(x)=\frac{x}{x^2+1},x>1[/imath]. Take the derivative and determine whether the function is decreasing or increasing for [imath]x>1[/imath].

 

[pka]

Alternating_series_test: If [imath]a_1\ge a_2\ge a_3\cdots a_n\ge \cdots\ge 0[/imath]
such that [imath]\left(a_n\right)\to 0[/imath] then the series [imath]\displaystyle\sum\limits_{n = 1}^\infty {{{( - 1)}^n}{a_n}} [/imath] converges.
In common parlance: if the sequence part of an alternating series is a decreasing to zero the series converges.
Can you explain why you think that does not apply here?

 

[Dr.Peterson]

The link you attached confirms what I applied in my attempt at the problem, however, I am still confused on why the Alternating Series Test failed and how to correct this.

You appear to be forgetting to take the absolute value of each term. You are using negative [imath]b_n[/imath]:


Note the absolute values here:

Alternating series test​

A series of the form [imath]\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}=a_{0}-a_{1}+a_{2}-a_{3}+\cdots[/imath] where either all a_n are positive or all a_n are negative, is called an alternating series.​

​

The alternating series test guarantees that an alternating series converges if the following two conditions are met:​

1. [imath]\displaystyle |a_{n}|[/imath] decreases monotonically, i.e., [imath]\displaystyle |a_{n+1}|\leq |a_{n}|[/imath], and
2. [imath]\displaystyle \lim _{n\to \infty }a_{n}=0[/imath]

 

[I Love Pi]

Alternating_series_test: If [imath]a_1\ge a_2\ge a_3\cdots a_n\ge \cdots\ge 0[/imath]
such that [imath]\left(a_n\right)\to 0[/imath] then the series [imath]\displaystyle\sum\limits_{n = 1}^\infty {{{( - 1)}^n}{a_n}} [/imath] converges.
In common parlance: if the sequence part of an alternating series is a decreasing to zero the series converges.
Can you explain why you think that does not apply here?

(Note: I attempted to prove the summation is conditionally convergent using the following theorem.)

I noticed that the the series is not (-1)^n , but (-1)^(n-1). Is this discrepancy why the Alternating Series Test does not apply? If the Alternating Series Test can’t be applied, how else could I prove the summation of An converges (and therefore prove the summation of An is conditionally convergent)?

 

[Dr.Peterson]

I noticed that the the series is not (-1)^n , but (-1)^(n-1). Is this discrepancy why the Alternating Series Test does not apply?

No, the alternating series test does apply! You are just applying it incorrectly because of the difference you point out.

Do you not see that you can just pull a -1 outside of the entire summation? You need to ensure that what you call b_n are positive (or else, as the Wikipedia statement of the test does, let them all be negative, as you did, but compare their absolute values).

​

See this lesson, which includes examples with different exponents on the (-1):

Calculus II - Alternating Series Test

In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. The Alternating Series Test can be used only if the terms of the series alternate in sign. A proof of the Alternating Series Test is also given.

tutorial.math.lamar.edu

Or you can show us the exact formulation of the test you were taught, and we can explain how you are misunderstanding it.

 

[I Love Pi]

That makes sense. Thank you very much, Dr. Peterson, pka, and Big Beach Banana! I greatly appreciate your help!