Problem: Evaluate the summation
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).
Then consider by simply multiplying each term by :
And now find the difference by subtracting the terms with equal denominators. We get
Notice that the numerator is now a polynomial of degree instead of . Repeating this, we have
Notice that the latter part is just a geometric series which sums to
so . QED.
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.
Practice Problem: Let be a polynomial with integer coefficients. Using the method of finite differences, predict the degree of .
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