Cauchy Condensation Test

Theorem
Let $\left \langle {a_n} \right \rangle: n \mapsto a\left({n}\right)$ be a decreasing sequence of strictly positive terms in $\R$ which converges with a limit of zero.

That is, for every $n$ in the domain of $\left \langle {a_n} \right \rangle$: $a_n > 0, a_{n+1} \le a_n$, and $a_n \to 0$ as $n \to +\infty$.

Then the series:


 * $\displaystyle \sum_{n=1}^\infty a_n$

converges iff the condensed series:


 * $\displaystyle \sum_{n=1}^\infty 2^n a\left({2^n}\right)$

converges.

Part 1
We will first show that if the condensed series $\displaystyle \sum_{n=1}^\infty 2^n a\left({2^n}\right)$ converges, then $\displaystyle \sum_{n=1}^\infty a_n$ converges as well.

Assume $\displaystyle \sum_{n=1}^\infty 2^n a\left({2^n}\right)$ converges.

Consider the graph of $a_n$ and the partial sums of $\sum 2^n a\left({2^n}\right)$:


 * Cauchycondensation1.png

The dotted black line represents the sequence $a_n$.

The $n$th rectangle has:


 * Base equal to $2^n$


 * Altitude equal to $a\left({2^n}\right)$

Hence the series of partial sums of the areas of the rectangles are:


 * $\displaystyle \sum_{n=1}^N 2^n a\left({2^n}\right)$

which is precisely the defined condensed series.

The series $\sum a_n$ can be viewed as the sum of the areas of rectangles with width $1$ and height $a_n$.

Hence the diagram suggests that the partial sums of $\sum a_n$ are not greater than the condensed partial sums:


 * $\displaystyle \sum_{n=1}^N a_n \le \sum_{n=1}^N 2^n a\left({2^n}\right)$

To formalize this claim, observe that because $\sum a_n$ is decreasing,


 * $a_1 + \underbrace{a_2 + a_3}_{\le 2a_2} + \underbrace{a_4 + a_5 + a_6 + a_7}_{\le 4a_4} + \cdots + a_N \le a_1 + 2a_2 + 4a_4 + \cdots + 2^N a\left({2^N}\right)$

As $n \to +\infty$, the RHS converges by hypothesis.

Hence $\displaystyle \sum_{n=1}^{\infty} a_n$ also converges, by the Comparison Test.

Part 2
We will show that if $\displaystyle \sum_{n=1}^\infty a_n$ converges, the condensed series $\displaystyle \sum_{n=1}^\infty 2^n a\left({2^n}\right)$ converges as well.

Assume $\displaystyle \sum_{n=1}^\infty a_n$ converges.

From the Combination Theorem for Sequences, $2\displaystyle \sum_{n=1}^\infty a_n = \displaystyle \sum_{n=1}^\infty 2a_n$ converges as well.

Consider, then, the graph of $a_n$ and of the partial sums of what will be shown to equal $\frac 1 2 \sum 2^{n} a\left({2^n}\right)$:


 * Cauchycondensation2.png

The dotted black line represents the sequence $a_n$.

The $n$th rectangle has:


 * Base equal to $2^n$


 * Altitude equal to $a\left( {2 \cdot 2^n }\right)$

Hence the series of partial sums of the areas of the rectangles are:


 * $\displaystyle \sum_{n=1}^N 2^n a\left(2 \cdot {2^n}\right)$

From Deletion of Terms from a Sequence, it is justified to examine only the behavior of $n$ for $n \ge 2$.

The series $\sum a_n$ can be viewed as the sum of the area of rectangles with height $a_n$ and width $1$.

Thus the diagram suggests that the partial sums of $\sum a_n$ satisfy the following inequality:


 * $\displaystyle \sum_{n=2}^N a_n \ge \frac 1 2 \sum_{n=2}^N 2^n a\left({2^n}\right)$


 * $\iff \displaystyle 2\sum_{n=2}^N a_n \ge \sum_{n=2}^N 2^n a\left({2^n}\right)$

To formalize this claim, observe that, as the sequence $a_n$ is decreasing,

By hypothesis, the LHS converges as $n \to +\infty$.

Hence the condensed series converges as well, by the Comparison Test.