The Comparison Test for Positive Series of Real Numbers

One extremely useful theorem for testing for the convergence of a series is called the comparison test for positive series. If we have two positive sequences, one of which is "larger" than the other, then if the corresponding larger series converges then the smaller series will also converge. Similarly, if the smaller series diverges then the larger series will also diverge.

Theorem 1: Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be positive sequences. Suppose that there exists an $N \in \mathbb{N}$ such that if $n \geq N$ we have that $a_n \leq b_n$. Then: a) If $\displaystyle{\sum_{k=1}^{\infty} b_k}$ converges then $\displaystyle{\sum_{k=1}^{\infty} a_k}$ converges. b) If $\displaystyle{\sum_{k=1}^{\infty} a_k}$ diverges to infinity then $\displaystyle{\sum_{k=1}^{\infty} b_k}$ diverges to infinity.

• Proof of a) Suppose that there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then:

(1)

\begin{align} \quad a_n \leq b_n \quad (*) \end{align}

• Consider the subseries $\displaystyle{\sum_{k=N}^{\infty} a_k}$ and $\displaystyle{\sum_{k=N}^{\infty} b_k}$. Since the full series $\displaystyle{\sum_{k=1}^{\infty} b_k}$ converges, say to some sum $s$ we see that:

(2)

\begin{align} \quad 0 < \sum_{k=N}^{\infty} a_k \leq \sum_{k=N}^{\infty} b_k < \sum_{k=1}^{\infty} b_k = s \end{align}

• So the series $\displaystyle{ \sum_{k=N}^{\infty} a_k}$ is bounded above and hence converges to some sum $s^*$. Furthermore, the finite series $\sum_{k=1}^{N-1} a_k$ also converges to a finite sum. Thus $\sum_{k=1}^{\infty} a_k$ converges.

• Proof of b) Statement (b) is the contrapositive of (a). $\blacksquare$