Question

find the convergence of the series 1/n(n+2)​

Answer 1

Step-by-step explanation:

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 :

sn = 1/3 + 1/8 + 1/15 + 1/24 + ... + 1 / n(n+2) + ...

I noticed that every term in sn 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 / n2

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.