Question

Show that the series $\sum _{k=1}^{\infty }\left(\frac{1}{k}-\frac{1}{k+2}\right)$is a telescoping series. Also, provide the general term $S_n$in its sequence of partial sums and find the sum of the series if it converges.

Short answer

The series $\sum _{k=1}^{\infty }\left(\frac{1}{k}-\frac{1}{k+2}\right)$has general term $S_n=\frac{3}{2}-\left(\frac{1}{n+1}+\frac{1}{n+2}\right)$and sum of the series is $\frac{3}{2}$.

Step-by-step solution

Step 1. Given information.

Given a series $\sum _{k=1}^{\infty }\left(\frac{1}{k}-\frac{1}{k+2}\right)$.

Step 2. Find if the series is telescoping series or not.

The series can be rewritten as follows.

$\sum _{k=1}^{\infty }\left(\frac{1}{k}-\frac{1}{k+2}\right)=\sum _{k=1}^{\infty }\left(\frac{1}{k}+\frac{1}{k+1}-\frac{1}{k+1}+\frac{1}{k+2}\right)$

Consider $a_k=\frac{1}{k}+\frac{1}{k+1}$.

It follows that $S_n=\sum _{n=1}^{\infty }\left(a_n-a_{n+1}\right)$.

Note that between each two consecutive pairs, the second term of a pair cancels with the first term of the subsequent pair.

The cancellation between these values gives rise to the term “telescoping” series in that each term in the sequence of partial sums collapses like a

folding telescope.

Now, $S_n=a_1-a_{n+1}$.

Therefore, the series is a telscoping series.

Step 3. Find the sum of the series if it converges.

Since $a_n=\frac{1}{n}+\frac{1}{n+1}$, it follows that as n tends to infinity, $a_n\to 0$.

So, as n tends to infinity, $S_n\to a_1$.

Note that $a_1=\frac{1}{1}+\frac{1}{2}$, that is $\frac{3}{2}$.

Therefore, sum of the series is $\frac{3}{2}$.