We get a lot of questions on partial fractions, on how to simplify them and how to solve questions that involve a summation of a series of fractional terms. In this post, I will solve 3 sums (in increasing order of difficulty) to explain the concept of partial fractions.

Try these questions on your own before you go to their solutions.

- What is

Solution:

Therefore, the sum can be expressed as:

This question becomes easy once we spot that

How does one ‘spot’ this?

First step, write down the general nth term of the expression. In the above equation,

Now, we start with the assumption that can be broken as two simpler fractions, say, like

Now, we need to find A and B.

Or, A + B = 0, A = 1. Or, A = 1, B = -1

Now, try another question of a similar type:

2. Find

Assuming that you have read and understood the concept behind the solution to the previous question, I am going to jump straight to the solution.

Each term can be expressed as:

After cancelling out the terms that appear as positive-negative pairs, we are left with:

Now, try another question of a similar type but with a different way of solving.

3. If A = and B = find the value of .

This must be approached in a different way. Notice that in A, the sum of the terms in the denominator is equal. So, the terms can be written as:

We need to find a way to simplify B as well. Since there are some negative terms, let us try to eliminate them by adding and subtracting

=>

=>

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