# 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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Integra-what? Topic: Calculus/S&S. December 1st, 2006

Problem: Write as an integral.

Solution: We first notice that

.

So our summation becomes

.

But we can switch the order of the integral with the two sums because . Hence we obtain

.

We remember, however, that the Taylor series expansion of , so substituting we have

.

Similarly, , so we replace one more time to get

.

QED.

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Comment: Transforming sums into integrals and vice versa are powerful techniques for evaluating these expressions. The numerical value turns out to be , but we’ll save that for another time…

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Practice Problem: (1978 Putnam – B2) Evaluate

.

### 4 Responses to “Integra-what? Topic: Calculus/S&S.”

1. t0rajir0u Says:

the inside sum telescopes, the outside sum is evaluated by using the asymptotic form of the harmonic series

2. amirhtlusa Says:

the solution that only involves partial fractions for the practice problem turns out to be quite ugly.

3. amirhtlusa Says:

looking at your integral, reminds me of the following quite nice problem:

evaluate the int_0^1 ln(1-x) ln(1+x) dx.

Yeah, I saw that on AoPS. The solution was a series expansion though… and then using the harmonic series, right?