The divergence test

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If an infinite sum converges, then its terms must tend to zero.

As one contemplates the behavior of series, a few facts become clear. In order to add an infinite list of nonzero numbers and get a finite result, “most” of those numbers must be “very near” $0$ . Think of this in the opposite sense: what happens if you try to sum $\sum _{n=1}^\infty 2$ ?

If a series diverges, it means that the sum of an infinite list of numbers is not finite (it may approach $\pm \infty$ or it may oscillate), and:

The series will still diverge if the first term is removed.
The series will still diverge if the first $10$ terms are removed.
The series will still diverge if the first $1000000$ terms are removed.
The series will still diverge if any finite number of terms from anywhere in the series are removed.

These concepts are very important and lie at the heart of the next theorems.

Divergence test Consider the series $\sum _{n=1}^\infty a_n.$

(a): If $\sum _{n=1}^\infty a_n$ converges, then $\lim _{n\to \infty }a_n =0$ .
(b): If $\lim _{n\to \infty }a_n \neq 0$ , then $\sum _{n=1}^\infty a_n$ diverges.

Note that the two statements above are really the same. In order to converge, the limit of the terms of the sequence must approach $0$ ; if they do not, the series will not converge.

This theorem does not state that if $\lim _{n\to \infty } a_n = 0$ then $\sum _{n=1}^\infty a_n$ converges.

The standard example of a sequence whose terms go to zero, and yet does not converge, is the harmonic series. The Harmonic sequence, $(1/n)$ , converges to $0$ while the Harmonic Series, $\sum _{n=1}^\infty \frac {1}{n}\qquad \text {diverges.}$

Let’s see if you’ve digested what we’ve been saying:

Which of the following statements are true? Mark all that apply.

If $\sum _{k=0}^\infty a_k$ is convergent, then $\lim _{k \to \infty } a_k = 0$ If $a_k \to 0$ as $k \to \infty$ , then $\sum _{k=0}^\infty a_k$ is convergent If $\sum _{k=0}^\infty a_k$ is divergent, then $\lim _{k \to \infty } a_k \neq 0$ If $\lim _{k \to \infty } a_k \neq 0$ , then $\sum _{k=0}^\infty a_k$ is divergent

We say that a series “passes the divergence test” if its sequence of terms tends to zero. Which of the following series pass the divergence test?

$\sum _{k=3}^\infty \frac {1}{\ln { k }}$ $\sum _{k=0}^\infty \sin (k)$ $\sum _{k=0}^\infty \frac {\sin (k)}{k^2}$ $\sum _{k=5}^\infty \frac {k+7}{k+6}$ $\sum _{k=0}^\infty \frac {2k}{k - 5}$

Restating this point again (because it is very important): passing the divergence test means that a series has a chance to converge. The divergence test cannot tell us whether a series converges.

Some questions

Suppose $(a_n)_{n \geq 1}$ is a sequence and $\sum ^{\infty }_{n= 1} a_n$ converges to $L>0$ . Let $S_n = \sum ^n_{k=1} a_k$ . Select all statements that must be true:

$\lim _{n \to \infty } a_n = L$ $\lim _{n \to \infty } a_n = 0$ $\lim _{n \to \infty } S_n = 0$ $\lim _{n \to \infty } S_n = L$ $\sum ^{\infty }_{n=1} S_n$ must diverge. $\sum ^{\infty }_{n=1} (a_n+1) = L+1$ The divergence test tells us $\sum ^{\infty }_{n= 1} a_n$ converges to $L$ .

Suppose that $\seq {a_n}_{n \geq 1}$ is a decreasing sequence. Let $S_n = \sum ^n_{k=1} a_k$ and suppose $\lim _{n \to \infty } S_n$ does not exist. Select all statements that must be true:

$\lim _{n \to \infty } a_n$ does not exist. $\sum ^{\infty }_{k=1} a_k$ could converge. $\sum ^{\infty }_{n=1} S_n$ must diverge. $\seq {S_n}$ must be monotonic. $\seq {S_n}$ must be bounded. $\lim _{n \to \infty } S_n = -\infty$ The divergence test applied to $\sum ^{\infty }_{k= 1} a_k$ would guarantee that $\sum ^{\infty }_{k=1} a_k$ diverges.

Suppose that $\seq {a_n}_{n \geq 1}$ is a sequence with $a_n > 0$ for all $n \geq 1$ . Let $S_n = \sum ^n_{k=1} a_k$ and suppose $\lim _{n \to \infty } S_n = L$ . Select all statements that must be true:

$\sum ^{\infty }_{k=1} a_k = L$ $\lim _{n \to \infty } a_n = 0$ $\seq {S_n}$ must be monotonic $\seq {S_n}$ must be bounded $\sum ^{\infty }_{n=1} (a_n-L) = 0$ $\sum ^{\infty }_{n=1} S_n$ must diverge The divergence test applied to $\sum ^{\infty }_{k= 1} a_k$ would guarantee that $\sum ^{\infty }_{k=1} a_k$ converges.

It’s a great idea at this point to stop and compare the previous two questions.