Convergence of series log(1-1/n^2)

  • #1
Felipe Lincoln
Gold Member
99
11

Homework Statement


Find the sum of ##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) ##

Homework Equations


No one.

The Attempt at a Solution


At first I though it as a telescopic serie:
##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) =\ln\left(\dfrac{3}{4}\right) + \ln\left(\dfrac{8}{9}\right) + \ln\left(\dfrac{15}{16}\right) + \dots < 0##
But it doesn't look like so.
we know from it's terms that ##s_n## is in the interval of ##(L, \ln\left(3/4\right)]##
 

Answers and Replies

  • #2
tnich
Homework Helper
1,048
336

Homework Statement


Find the sum of ##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) ##

Homework Equations


No one.

The Attempt at a Solution


At first I though it as a telescopic serie:
##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) =\ln\left(\dfrac{3}{4}\right) + \ln\left(\dfrac{8}{9}\right) + \ln\left(\dfrac{15}{16}\right) + \dots < 0##
But it doesn't look like so.
we know from it's terms that ##s_n## is in the interval of ##(L, \ln\left(3/4\right)]##
You are right about its being a telescoping series. How can you factor ##1-\frac 1 {n^2}##?
 
Last edited:
  • #3
Delta2
Homework Helper
Insights Author
Gold Member
3,562
1,375
Using basic property of logarithms you can find that it is equal to
$$\ln\prod_{n=2}^{\infty} \frac{(n-1)(n+1)}{n^2}$$

So you can work with that product instead and prove that the product is equal to ##\frac{1}{2}## so the limit is ##\ln\frac{1}{2}=-\ln2##

I know that usually is not a good idea to work with products but in this case it works..

Now that I see it again, this product is the product of two "telescopic" products ##\prod\frac{n-1}{n},\prod\frac{n+1}{n}##, so I guess the original series must be a sum of two telescopic series.
 
Last edited:

Related Threads on Convergence of series log(1-1/n^2)

Replies
6
Views
25K
Replies
5
Views
2K
Replies
16
Views
21K
  • Last Post
Replies
7
Views
3K
Replies
6
Views
7K
Replies
5
Views
8K
Replies
3
Views
6K
Replies
8
Views
1K
Replies
2
Views
1K
  • Last Post
Replies
1
Views
2K
Top