Hi,
I'm just about to hit my head on a table or a wall...
I have to determine if the series (sum symbol) (n=1 to infinity) 1 / n(n+2) is convergent or not, and if it is, what is its limit. In the book, the answer is "3/4".
So I began writing its partial sum :
s
_{n} = 1/3 + 1/8 + 1/15 + 1/24 + ... + 1 / n(n+2) + ...
I noticed that every term in s
_{n} is multiplied by : n(n+2) / (n+1)(n+2+1) = n(n+2) / (n+1)(n+3)
So I thought it was a geometric series with
a = 1 / a(n+2)
r = n(n+2) / (n+1)(n+3)
Since r will always be < 1 for every n >= 1, I tried to resolve a / (1 - r), but I never got 3/4.
Thinking I had to set a equal to "1/3", I did not get 3/4 either.
Then, I told myself it was a combinaison of two series :
1. 1 / n
^{2}
2. 1 / 2n
But the second one does not have a limit since it is 1 / 2 multiplied by the harmonic series (1 / n) which does not converge.
What am I doing wrong ? What did I not see ?
I just started this calculus class in university and my college maths are soooooooo far away (11 years).
Thanks for your help