Finding an explicit formula for the sequence of partial sums

[RJLiberator]

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)

 

[Stephen Tashi]

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

Yes.

If the first index is 1 (as it is in you problem) , the "n-th partial sum" of a series $\{a_n\}$ is defined to be $\sum_{i=1}^n a_n$. For example, the 2nd-partial sum of the series in your example is (1- 1/2) + (1/2 - 1/3), so you can check your formula agrees with that number when n = 2.

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

What do you mean by "this"? Your problem involves 3 different things that could converge or diverge.
1) The sequence
2) The sequence of partial sums of the above sequence
3) The infinite series defined by summing the sequence in 1) above.

The limit of a sequence $\{a_n\}$whose terms are given by a formula $a_n = f(n)$ is $lim_{n \rightarrow \infty} f(n)$.

The limit of a series $\sum_{i=1}^\infty a_n$ is given by the limit of its n-th partial sum as n approaches infinity. (Some textbooks use that statement as the definition for the limit of an infinite series.)

 

[RJLiberator]

Ah, the partial sum formula just clicked for me with that confirmation. I can see that works out.

As for convergence, I am learning to take the limit of the partial sum, this would converge to 1.