comparison test

Comparison test

Let ${\sum }_{k=1}^{\infty }{a}_k$ and ${\sum }_{k=1}^{\infty }{b}_k$ be series.
i) If $\sum b_k$ is absolutely convergent, and $|a_k|\le |b_k|\forall k\in \mathbb{N}$. Then, $\sum a_k$ is also absolutely convergent.
ii) If $\sum b_k=\infty$, and $0\le b_k\le a_k\forall k\in \mathbb{N}$, then also $\sum a_k=\infty$.

In other words: If all terms of one series are smaller than the corresponding terms of an absolutely convergent series, then the first series is also absolutely convergent. If all terms of one series are larger than the corresponding terms of a divergent series, then the first series is also divergent.

EXAMPLE

$$\sum _{k=1}^{\infty }|\frac{k^2+4k^2-3}{k^5+k+1}|\le \sum _{k=1}^{\infty }\frac{k^3+4k^3}{k^5}=5\sum _{k=1}^{\infty }\frac{1}{k^2}\le \infty$$

EXAMPLE

$$\sum _{k=1}^{\infty }|\frac{k^2+4k^2-3}{k^4-k+1}|\ge \sum _{k=1}^{\infty }\frac{k^3}{k^4}=\sum _{k=1}^{\infty }\frac{1}{k}=\infty$$

What does the comparison test state?

If all terms of one series are smaller than the corresponding terms of an absolutely convergent series, then the first series is also absolutely convergent. If all terms of one series are larger than the corresponding terms of a divergent series, then the first series is also divergent.