Consider the series ∞Σn=1 1/n(n+1) Write the series using sigma notation. and

Question

Write down the first four terms in the sequence of partial sums, and then find the exact formula for the N-th partial sum.

Use the exact formula to determine if the series converges or diverges. If it converges, find the value it converges to.

Answer 1

Answer 2

The first 4 terms are: 1/2, 1/6, 1/12, 1/20.

I will leave computation of the partial sums to you.

Since the terms are all less than 1/n² (which converges), the series converges.

I am not sure that the sum is known precisely. What I can tell you is that the value to which it converges is between 0 and π²/6 which is the sum of 1/n².

OK, I just checked.

It converges to 1.

i/[n(n+1)] = (1/n)-(1/n+1)

The sum "telescopes" leaving only one term which clearly goes to 1 as n gets large.