Question

Sequences: Prove the sum from n=1 to infinity of 1/(n(n+1))=1...

Answer 1

write as a telescoping sum

Answer 2

Hint: \( \large \frac{1}{n (1+n)} = \frac1n - \frac1{(n+1)} \)

Answer 3

The partial sum would be \[ 1-\frac{1}{1+n } \]

now if you take the limit to infinity you will get your desired result.

Answer 4

\[\sum_{n=1}^{\infty} 1/(n(n+1)) =1\]

Know that \[\sum_{k=1}^{\infty} 1/(k(k+1)) = \sum_{k=1}^{\infty} [1/k - 1/(k+1)]\]

Okay - this is real analysis and I haven't had Calculus in about 6 years. So I just take lim (n-> infty) of the sum?

Answer 5

I'm slow typing...thanks for the help.

Answer 6

Glad to help :)