# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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Square Sum Stuff. Topic: Polynomials/S&S. Level: AMC/AIME. October 27th, 2006

Problem: Evaluate the summation

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Solution: Here’s a technique that will help you evaluate infinite series that are of the form polynomial over exponential. It’s based on the idea of finite differences:

If is a polynomial with integer coefficients of degree then

is a polynomial of degree (not hard to show; just think about it).

So let

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Then consider by simply multiplying each term by :

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And now find the difference by subtracting the terms with equal denominators. We get

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Notice that the numerator is now a polynomial of degree instead of . Repeating this, we have

and

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Notice that the latter part is just a geometric series which sums to

so . QED.

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Comment: The method of finite differences is extremely useful and is basically a simplified version of calculus – in a very approximating sense. It’s a good thing to know, though, because then you have a better understanding of how polynomials work.

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Practice Problem: Let be a polynomial with integer coefficients. Using the method of finite differences, predict the degree of .

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### 2 Responses to “Square Sum Stuff. Topic: Polynomials/S&S. Level: AMC/AIME.”

1. t0rajir0u Says:

The coefficients don’t have to be integer…