# 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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Condensation Sensation. Topic: Calculus/S&S. March 27th, 2007

Theorem: (Cauchy Condensation Test) If is a monotonically decreasing sequence of positive reals and is a positive integer, then

converges if and only if converges.

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Problem: Determine the convergence of , where is a positive real.

Solution: Well, let’s apply the Cauchy condensation test. Then we know that

converges if and only if

does. But this clearly diverges due to the fact that the numerator is exponential and the denominator is a power function. QED.

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Comment: This is a pretty powerful test for convergence, at least in the situations in which it can be applied. The non-calculus proof for the divergence of the harmonic series is very similar to the Cauchy condensation test; in fact, the condensation test would state that

converges iff converges,

which clearly shows that the harmonic series diverges.

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Practice Problem: Determine the convergence of .

### 5 Responses to “Condensation Sensation. Topic: Calculus/S&S.”

1. Xuan Says:

u spelled “iff” wrong

iff = if and only if

3. t0rajir0u Says:

Hmm. I was going to ask for a proof, but it occurred to me that (condensed series)

4. t0rajir0u Says:

Oops. I wanted to write (condensed series) ≤ (original series) ≤ p * (condensed series). Yeah.

Anyway, Cauchy condensation gives

$$\displaystyle \sum_{n=2}^{\infty} \frac{p^n}{(n \ln p)^{n \ln p} }$$

Which clearly converges by ratio test.