c) Prove, that the series $\sum_{n=1}^{\infty}\frac{1}{(n+1)(n+2) }$ converges and calculate the value of the series
Hint: Telescoping sum.

Question

c) Prove, that the series $\sum_{n=1}^{\infty}\frac{1}{(n+1)(n+2)	}$ converges and calculate the value of the series
Hint: Telescoping sum.

anonymous · Answer

$$
\frac{1}{(n+1)
   (n+2)}=\frac{1}{n+1}-\frac{
   1}{n+2}\

s_n=\ \frac{1}{2}-\frac 1 3+\

 \frac{1}{3}-\frac 1 4+\
 \frac{1}{4}-\frac 1 4+\
\cdots\
\cdots\

\frac{1}{n+1}-\frac{
   1}{n+2}\

s_n =\frac 1 2 -\frac{
   1}{n+2}\

\lim_{n\to \infty} s_n=\frac 1 2 

$$

anonymous · Answer

thank you very much dear mr Elias, there is some parts that i dont understand but i will ask later, i must first ensure points for tomorow :)

anonymous · Answer

$$
\sum _{n=1}^{\infty }
   \frac{1}{(n+1)
   (n+2)}=\frac{1}{2}
$$

anonymous · Answer

and its not convergence right ?
its divergence

anonymous · Answer

No. It is convergent and its sum is 1/2

anonymous · Answer

hm ok.. thanks