Sequence of Terms Divergence Criterion for Infinite Series

Recall from the Convergence and Divergence of Infinite Series page that an infinite series $\displaystyle{\sum_{k=1}^{\infty} a_k}$ is said to converge to the sum $s$ if the corresponding sequence of partial sums $(s_k)_{k=1}^{\infty}$ converges to $s$, and the series is said to diverge if it does not converge to any sum $s$.

We will now look at a very important theorem which provides us with divergence criterion for a series.

Theorem 1: If the series $\displaystyle{\sum_{k=1}^{\infty} a_k}$ converges then $\displaystyle{\lim_{k \to \infty} a_k = 0}$.

• Proof: Suppose that $\sum_{k=1}^{\infty} a_k = s$. Let $(s_k)_{k=1}^{\infty}$ denote the sequence of partial sums of this sequence. Then we have that:

(1)

\begin{align} \quad s_k &= a_k - s_{k-1} \\ \quad a_k &= s_k - s_{k-1} \end{align}

• Taking the limit as $k \to \infty$ from both sides and we see that:

(2)

\begin{align} \quad \lim_{k \to \infty} a_k &= \lim_{k \to \infty} s_k - s_{k-1} \\ \quad \lim_{k \to \infty} a_k &= \lim_{k \to \infty} s_k - \lim_{k \to \infty} s_{k-1} \\ \quad \lim_{k \to \infty} a_k &= s - s \\ \quad \lim_{k \to \infty} a_k &= 0 \quad \blacksquare \end{align}

Corollary 1: If $\displaystyle{\lim_{k \to \infty} a_k \neq 0}$ then the series $\displaystyle{\sum_{k=1}^{\infty} a_k}$ diverges.

• Proof: This is simply the contrapositive statement of Theorem 1. $\blacksquare$

The corollary above is much more applicable than Theorem 1 in general. It gives us criterion for quickly determining whether many series are divergent. Note that the converse of Theorem 1 is not true in general though, that is, if $\lim_{k \to \infty} a_k = 0$ then we cannot necessarily determine whether the series $\displaystyle{\sum_{k=1}^{\infty} a_k}$ converges or diverges.