# Can you explain briefly about the Telescoping Summation?

lfryerda | Certified Educator

A telescoping sum or series is a series where every term except the first few and the last few terms cancel out.  This means that the series becomes very easy to turn into a closed expression.

It is not usually immediately obvious that a series will telescope, and usually some manipulation with partial fractions or other algebraic techniques may be necessary to make it a telescoping sum.

For example, consider the sum

sum_{k=1}^n 2/{k(k+2)}

Although it isn't a telescoping sum yet, we can use partial fractions on this sum.  That is, break the terms into two fractions:

2/{k(k+2)}=a/k+b/{k+2}

and by finding common denominators on the RHS, we see that a=1 and b=-1.

This means that the sum now becomes

\sum_{k=1}^n(1/k-1/{k+2})

=(1-1/3)+(1/2-1/4)+(1/3-1/5)+\cdots+(1/n-1/{n+2})

and rearranging the terms gives

=1+1/2+(1/3-1/3)+(1/4-1/4)+\cdots+(1/n-1/n)-1/{n+1}-1/{n+2}

and every term cancels out except the first two and the last two, so the final sum is just

=1+1/2-1/{n+1}-1/{n+2}

so this means that

\sum_{k=1}^n 2/{k(k+2)}={n(3n+5)}/{2(n+1)(n+2)}