Converge of infinite series

In summary, the conversation discusses how to show that the series Σ_{n=1}^\infty ln(1 -1/n^2) is equal to -ln 2. It is suggested to use telescoping sums and to make use of the expression ln(a*b)=lna + lnb. By starting the sum at n=2, it indirectly proves that the product of (n^2-1)/n^2 from n=2 to infinity is equal to 1/2.
  • #1
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Show that Σ[tex]_{n=1}^\infty ln(1 -1/n^2) = -ln 2[/tex]

I'm not sure how to do this. Should I use telescopic sums or should I make a function [tex]y = ln(1 - 1/n^2)[/tex]? Is it possible to use telescopic sums here?
 
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  • #2
When I make [tex]y = ln(1 - 1/n^2)[/tex]..then the limit of y = 0 which yields no conclusion about the series.
 
  • #3
The sum cannot start at n=1 because then the first term blows up. I get the correct answer if n starts at 2. Two hints:

1) Yes it's a telescoping series problem.

2) ln(a+b) is not a friendly expressiom but ln(a*b) is because ln(a*b)=lna + lnb. And notice that [tex]1-\frac{1}{n^2}=\frac{n^2-1}{n^2}=\frac{(n-1)(n+1)}{n^2}[/tex]
 
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  • #4
Incidentally, this problem proves indirectly that

[tex]\prod_{i=2}^{\infty}\frac{i^2-1}{i^2}=\left(\frac{3}{4}\right)\left(\frac{8}{9}\right)\left(\frac{15}{16}\right)\left(\frac{24}{25}\right)...=\frac{1}{2}[/tex]

Nice!
 
  • #5
Thank you!
 

1. What is the definition of "convergence" in the context of infinite series?

Convergence in the context of infinite series refers to the behavior of a series as the number of terms in the series approaches infinity. If the sum of the terms in the series approaches a finite value as the number of terms increases, the series is said to be convergent.

2. How do you determine if an infinite series converges or diverges?

There are several tests that can be used to determine the convergence or divergence of an infinite series, including the ratio test, the root test, and the comparison test. These tests involve evaluating the behavior of the terms in the series and comparing them to known convergent or divergent series.

3. Are there different types of convergence for infinite series?

Yes, there are two main types of convergence for infinite series: absolute convergence and conditional convergence. A series is said to be absolutely convergent if the sum of the absolute values of its terms converges. A series is conditionally convergent if it converges, but not absolutely.

4. Can an infinite series converge to more than one value?

No, an infinite series can only converge to a single value. If the sum of the terms in a series approaches different values as the number of terms increases, the series is said to be divergent.

5. What is the importance of understanding convergence in infinite series?

Understanding convergence in infinite series is crucial in many areas of mathematics and science. It allows us to make accurate predictions and calculations in fields such as physics, engineering, and statistics. Additionally, understanding the convergence of series is essential for developing new mathematical techniques and solving complex problems.

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