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This essential trains for: AMC-10, AMC-12, AIME.
Partial fraction decomposition is a technique used in summation and integration.
The purpose is to break down a term expressed as an algebraic fraction into simpler terms. The simpler terms may then exhibit a pattern of cancellation among successive terms that allows us to telescope the sum. In the case of integration, it helps distribute the integral into two simpler integrals.
Example: Compute the sum
First, notice the pattern that yields the sequence of denominators:
The general term of this sum has the form:
Let us try to write this term as a sum of simpler algebraic fractions:
Perform the addition of algebraic fractions:
Find A and B by identifying the polynomials that result at the numerator:
and the decomposition in partial fractions is:
Now the sum can be written in the form:
The pattern of term cancellations is very obvious. The sum results:
However, the pattern of term cancellation is not always as simple and it is always a good practice to write methodically a few terms from the start and a few terms from the end and use accuracy in determining which terms exactly cancel out.
In a problem solving context, the denominator may be expressed as a polynomial expression that needs to be factored out before attempting the decomposition. As in:
where the degree n of the polynomial at the numerator is lower than the degree m of the polynomial at the denominator.
Since a polynomial can always be factored into linear and at most quadratic factors, we can attempt a decomposition in simple fractions with numerators that are constant or at most linear polynomials.
We notice that the factors of the denominator cannot be factored into linear terms with real coefficients since the discriminants of the two quadratic expressions are negative.
The numerators of the simple fractions can be at most linear.
Assume the decomposition:
and proceed to identify the polynomials at the numerators:
with the solution:
Finally, the decomposition is:
And now what?
We still have to make the crucial observation that:
which helps us to finally telescope the sum:
After canceling the terms: