Cauchy_Condensation_Test

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Definition

Suppose $(b_n)$ is decreasing and satisfies $b_n \geq 0$ for all $n \in \mathbb{N}$. Then the series $\sum_{n=1}^{\infty} b_n$ converges if and only if the series $\sum_{n=0}^{\infty} 2^n b_{2^n}$ converges.

Proof

Group terms: $b_1 + (b_2 + b_3) + (b_4 + b_5 + b_6 + b_7) + \cdots$

Since $(b_n)$ is decreasing:

• $b_2 + b_3 \leq 2b_2$
• $b_4 + b_5 + b_6 + b_7 \leq 4b_4$

Thus $\sum b_n \leq b_1 + \sum_{n=1}^{\infty} 2^n b_{2^n}$.

Similarly, $b_2 + b_3 \geq 2b_4$, etc., giving $\sum b_n \geq \frac{1}{2}\sum 2^n b_{2^n}$.

By the Comparison_Test, the two series converge or diverge together.