- #1

RJLiberator

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## Homework Statement

I am trying to wrap my head around what it means to find an explicit formula for the sequence of partial sums.

Question: Find an explicit formula for the sequence of partial sums and determine if the series converges.

a) sum from n=1 to n=infinity of 1/(n(n+1))

## Homework Equations

## The Attempt at a Solution

This is a telescoping series.

We can reformat the sum as follows:

a) sum from n=1 to n=infinity of (1/n - 1/(n+1))

Writing out the first few terms we see

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

Clearly, everything cancels out except 1 and 1/(n+1).

The question asks for an explicit formula for the sequence of partial sums. Would that simply be (1-1/(n+1)) ?

Additionally, how can I tell that this converges (my hunch is that it does, since the terms are telescoping and canceling each other out)