Posted by **Hannah** on Saturday, July 11, 2009 at 7:45pm.

Evaluate sum(n=2,n=infinity) ln(1-1/n^2) if it exists.

- Math -
**Count Iblis**, Saturday, July 11, 2009 at 8:27pm
1-1/n^2 = (n^2 - 1)/n^2 =(n+1)(n-1)/n^2

------>

Log(1-1/n^2) =

Log(n+1) + Log(n-1) - 2 Log(n).

If we define the difference operator Delta as:

(Delta f)(n) = f(n+1) - f(n)

Then applying Delta twice gives:

(Delta^2 f)(n) =

f(n+2) - 2f(n+1) + f(n)

The formula for Log(1-1/n^2) has the same structure. We see that:

Log(1-1/n^2) = Delta^2 f(n)

with:

f(n) = log(n-1)

In general, we have:

Sum from n = r to s of (Delta g)(n) =

g(r+1) - g(r) + g(r+2) - g(r+1) + ...

+ g(s) - g(s-1) + g(s+1) - g(s) =

g(s+1) - g(r)

So, we have:

Sum from n = 2 to N of Log(1-1/n^2) =

Sum from n = 2 to N of

Delta(Delta f)(n) =

(Delta f)(N+1) - (Delta f)(2)

Now

f(n) = log(n-1),

so

(Delta f)(n) = Log(n) - log(n-1) =

Log[n/(n-1)]

and we see that:

(Delta f)(N+1) - (Delta f)(2) =

Log[(N+1)/N] - Log(2)

The sum to infinity is the limit for N to infinity. We can write:

Log[(N+1)/N] = Log(1+1/N) =

1/N + O(1/N^2)

So, this terms goes to zero in the limit N to infinity.

The summation is thus equal to -Log(2)

## Answer This Question

## Related Questions

- Calculus - Determine convergence or divergence for the following series. State ...
- calculus - find the sum of the series: 1. the sum from n=1 to infinity of ((-1)^...
- Math - The sum to infinity of a G.P. is 16 and the sum of the first 4terms is 15...
- math - Evaluate the sum: [(infinity) sigma i=0] (i/(4^i)) the answer is 4/9, but...
- calculus - Recall that a function G(x) has the limit L as x tends to infinity, ...
- MATH - I have been trying to do this problem for a couple of days but i cant ...
- Calculus (integrals) - Use the following formula for the sum of the cubes of the...
- calculus - Recall that a function G(x) has the limit L as x tends to infinity, ...
- CALC 2 - a. Consider the following limit as a fact: lim n-> infinity ((n!)^1/...
- Calculus - Solve the following initial value problem and determine the interval ...

More Related Questions