Expert: Paul Klarreich - 3/11/2006

Question
I must find the sum of the series...
sum from n=2 to inifnity of ln(1-(1/(n^2))).

I have NO idea how to even start it... I tried taking the limit of Sn as it approached infinity, but that gave me 0, so I would think that's impossible... and I have no idea if I should do anything different when n does not equal 1, because n=2 here...

Any help is appreciated.

Answer
Hi, Jose,

Question:  I must find the sum of the series...
sum from n=2 to inifnity of ln(1-(1/(n^2))).

I have NO idea how to even start it... I tried taking the limit of Sn as it approached infinity, but that gave me 0, so I would think that's impossible... and I have no idea if I should do anything different when n does not equal 1, because n=2 here...

Any help is appreciated.



--------------------------

If you thought that when each term of the series converges to zero that the sum must converge, all I can say is 'No way, Jose.'


Sorry, I just couldn't resist.  Of course,  a(n) -> 0 is a necessary condition for convergence, but not sufficient.

WARNING: THE MATERIAL BELOW MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  BE SURE TO VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.


You have this sum to test and evaluate:

               1
sum(..) ln(1 - ---)
              n^2

Combine the fractions.

          n^2 - 1
sum(..) ln(-------)
            n^2

Factor on top:

         (n + 1)(n - 1)
sum(..) ln-------------
               n^2

Use logarithm properties:

sum(..) [ln(n+1) + ln(n-1) - 2 ln n]

For n = 2 to infinity, this looks like:

[ln 3  +  ln  1  - 2 ln 2] +
[ln 4  +  ln  2  - 2 ln 3] +
[ln 5  +  ln  3  - 2 ln 4] +
[ln 6  +  ln  4  - 2 ln 5] +
[ln 7  +  ln  5  - 2 ln 6] +
[ln 8  +  ln  6  - 2 ln 7] +
....

Which, after the middle terms, except the first, get killed by half the third terms of the previous line, looks like this:

[ln 3 + ln 1 -   ln 2] +
[ln 4        -   ln 3] +
[ln 5        -   ln 4] +
[ln 6        -   ln 5] +
[ln 7        -   ln 6] +
[ln 8        -   ln 7] + ...

which, after the first terms get killed by the third terms of the following line, looks like this:


[     + ln 1 -   ln 2] +
[                    ] +
[                    ] +
[                    ] +
[                    ] +
[                    ] + ...

After the carnage, all that is left is:

= ln 1 - ln 2 = 0 - ln 2 = - ln 2

In case you don't believe this (I didn't either, for a while) you can try setting up a calculation (Excel is good for this).  Mine confirmed the convergence nicely.