# Sum of Telescoping Series

Another fascinating series is telescoping series. This series is quite different from a regular progression. If you came across to a problem that ask for the sum and you can’t find it by using arithmetic or geometric series. It could be the solve using telescoping technique. This series requires our knowledge to express fractional expression to its partial fraction form. I won’t be surprise if this is your first time to hear such series because the application of this topic is usually for Olympiad Mathematicians.

Ingenious way to express telescopic sum

**Sample Problem 1:**

Find the sum of the series

**Solution:**

We can express the nth term as

As mentioned above

Let n=1

Let n=2

Let n=3

.

.

.

Let n=2013

Expressing to its equivalent partial fractions we have

Cancel out terms leaving

Therefore we can conclude that

**Sample Problem 2:**

Find the sum of

**Solution:**

Using the law of logarithm,

Let k=1

Let k=2

Let k=3

.

.

.

Let k=99

Cancel out terms from log2 to log99 we have

but log1=0

Therefore we can conclude that the formula for Find the sum of

**Sample Problem 3:**

Find the sum of

**Solution:**

The nth term of this series is

By rationalizing the denominator,

Factor the common terms

This means that

Let n=1

Let n=2

Let n=3

.

.

.

Let n=840

We can express

As

Cancel out terms leaving