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Study Tips: Sum of Infinite Telescoping Series

This tip on sequences and series was contributed by one of our top tutors, Rushil Shah.

    What you will learn from this tip:

  • An understanding of a telescoping series
  • The trick to calculate the sum of a telescoping series

Telescoping series

A telescoping series is any series where nearly every term cancels with a preceding or following term.
An example is the following series:

    • The general method to tackle such a problem is

    • To try adding and subtracting a number from the numerator


  • To write the general term of the series and then manipulate it into the telescoping form

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For the example shown above, we will add and subtract 1 from the numerator to simplify it.

So simple right? Here is another tip. A telescoping series can be checked by replacing the term ‘r’ by ‘r+1’ in the first term. If you do so, and get the second term, the series is telescoping.

Here is another example.

Consider the sequence given below

First, we write the general term as

where r = 1, 2, 3, …

The trick here is to subtract any two consecutive terms in the denominator, and then to place that term in the numerator.
NOTE: This is only applicable if the denominator is a product of numbers in increasing order.

The above series is telescoping. Thus it’s sum equal to 1. Therefore,

That wasn’t so hard, was it? Stay tuned for the next edition of Study Tips by HashLearn!

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