Posted by **Sean** on Thursday, May 28, 2009 at 12:28pm.

What does the following infinite series starting at k=2 converge to: Σ ln (1 - 1/k^2)

In other words, what does this converge to: ln(1 - 1/4) + ln(1 - 1/9) + ln(1 - 1/16) + ln(1 - 1/25) + ln(1 - 1/36) + ...

I assume the first step is this:

Σ ln (1 - 1/k^2) = ln Π (1 - 1/k^2)

But from there, I don't know how to convert this to closed form and continue.

- Math -
**Count Iblis**, Thursday, May 28, 2009 at 12:59pm
The kth factor in the product is:

1-1/k^2 = (k^2-1)/k^2 =

(k+1)(k-1)/k^2

We can write this as:

f(k+1)/f(k)

where

f(k) = k/(k-1)

So, then we ave:

f(k+1)/f(k)=

(k+1)/k * (k-1)/k

which is exactly the kth term.

The product can then be written as:

[f(3)/f(2)]*[f(4)/f(3)]*[f(5)/f(4)]*...

= 1/f(2) as all the other factors cancel.

- Math -
**Count Iblis**, Thursday, May 28, 2009 at 1:05pm
Note that this is a special case of the formula:

sin(pi x)/(pi x) =

Product from k = 1 to infinity of

[1 - x^2/k^2]

- Math -
**Sean**, Thursday, May 28, 2009 at 1:28pm
Ah, thanks! Makes perfect sense

## Answer This Question

## Related Questions

- Math - Where does this infinite series converge: Sigma (k = 1 to infinity): 1/(...
- Calculus 2 - n=1 series to infinity (-5^n)/n^3 does it absolutely converge, ...
- math - for what values of x does the series converge absolutely and for what ...
- Calculus - Infinite series question. an=2+(0.99)^n Converge or Diverge?
- algebra - does the infinite geometric series diverge or converge? 1/7 + 1/14 + 1...
- Algebra 2 - I need steps on how to complete this please i am so confused and ...
- Math - formalize the following statment: a_n is a sequence that doesnt converge ...
- CALCULUS-URGENT- no one will respond!!! - we know the series from n=0 to ...
- probability - For each of the following sequences, determine the value to which ...
- Probability - For each of the following sequences, determine the value to which ...

More Related Questions