Infinite sum of 1/(n(n+1))?

Question

Hello, it is a simple question...
I can't find the way to go to the solution, but I already know it is 1.
I find that 1/(n(n+1)) = 1/n - 1/(n+1) and then?
Any suggestions should be very usefull.
Thanks

Answer 1

1/(n(n+1)) = 1/n - 1/(n+1)

Writing the series for simplification:
Start putting values of n from n = 1 ,n=2...............

=1/1 - 1/(1+1) + 1/2 - 1(2+1) + 1/3 - 1/(3+1) +...-1/n + 1/n -1/(n+1)

=1 - 1/2 + 1/2 -1/3 + 1/3 - 1/4 + 1/4 - ......-1/n + 1/n -1/(n+1)

You see that the adjacent terms get cancelled!

= 1 - 1/(n+1)

Since n tends to infinity 1/(n+1) = 0

Thus answer is 1

Answer 2

it is easier in textbook examples, please check