# Mathematical Food for Thought

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

• ## Meta

eek! Topic: Calculus/S&S. Level: AIME/Olympiad. December 9th, 2006

Problem: (Problem-Solving Through Problems – 5.4.15) Let and for . Show that

.

Solution: Well, the LHS looks suspiciously like a Taylor series, so maybe we can find a function. A good choice is . Notice that . Testing a few derivatives, we hypothesize that

.

By induction, we easily have

,

which exactly satisfies

so we indeed have . Then and the Taylor series of centered at zero is

.

Hence our desired sum is the above series evaluated at , which is simply

.

QED.

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Comment: A neat problem resulting from Taylor series. They are useful in all sorts of ways, especially evaluating other series. If we can reduce a given series to the Taylor series of a function like we did in this problem, evaluating it becomes plugging in a point.

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Practice Problem: (Problem-Solving Through Problems – 5.4.17) Show that the functional equation

is satisfied by

with .

### 2 Responses to “eek! Topic: Calculus/S&S. Level: AIME/Olympiad.”

1. t0rajir0u Says:

Well, x f(ix) = sin x but plugging that in doesn’t make it very nice at all

Can’t think of a good way to do this. I tried a trig substitution x = tan t/2 but even that wasn’t very effective.