$\newenvironment {prompt}{}{} \newcommand {\ungraded }{} \newcommand {\todo }{} \newcommand {\oiint }{{\large \bigcirc }\kern -1.56em\iint } \newcommand {\mooculus }{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}} \newcommand {\npnoround }{\nprounddigits {-1}} \newcommand {\npnoroundexp }{\nproundexpdigits {-1}} \newcommand {\npunitcommand }{\ensuremath {\mathrm {#1}}} \newcommand {\RR }{\mathbb R} \newcommand {\R }{\mathbb R} \newcommand {\N }{\mathbb N} \newcommand {\Z }{\mathbb Z} \newcommand {\sagemath }{\textsf {SageMath}} \newcommand {\d }{\mathop {}\!d} \newcommand {\l }{\ell } \newcommand {\ddx }{\frac {d}{\d x}} \newcommand {\zeroOverZero }{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }{\textbf } \newcommand {\unit }{\mathop {}\!\mathrm } \newcommand {\eval }{\bigg [ #1 \bigg ]} \newcommand {\seq }{\left ( #1 \right )} \newcommand {\epsilon }{\varepsilon } \newcommand {\phi }{\varphi } \newcommand {\iff }{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\arrowvec }{{\overset {\rightharpoonup }{#1}}} \newcommand {\vec }{{\overset {\boldsymbol {\rightharpoonup }}{\mathbf {#1}}}\hspace {0in}} \newcommand {\point }{\left (#1\right )} \newcommand {\pt }{\mathbf {#1}} \newcommand {\Lim }{\lim _{\point {#1} \to \point {#2}}} \DeclareMathOperator {\proj }{\mathbf {proj}} \newcommand {\veci }{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }{\vec {\boldsymbol {\l }}} \newcommand {\uvec }{\mathbf {\hat {#1}}} \newcommand {\utan }{\mathbf {\hat {t}}} \newcommand {\unormal }{\mathbf {\hat {n}}} \newcommand {\ubinormal }{\mathbf {\hat {b}}} \newcommand {\dotp }{\bullet } \newcommand {\cross }{\boldsymbol \times } \newcommand {\grad }{\boldsymbol \nabla } \newcommand {\divergence }{\grad \dotp } \newcommand {\curl }{\grad \cross } \newcommand {\lto }{\mathop {\longrightarrow \,}\limits } \newcommand {\bar }{\overline } \newcommand {\surfaceColor }{violet} \newcommand {\surfaceColorTwo }{redyellow} \newcommand {\sliceColor }{greenyellow} \newcommand {\vector }{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }{} \newcommand {\HyperFirstAtBeginDocument }{\AtBeginDocument }$

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

In order to determine if a series $\sum _{k=1}^{\infty } a_k$ converges, we took the following approach.
• Consider the associated sequence $\{s_n\}$ of partial sums, where $s_n=\sum _{k=1}^n a_k$.
• Try to find an explicit formula for the term $s_n$. If you can find such a formula, analyze $\lim _{n \to \infty s_n}$.
• If the limit exists, $\sum _{k=k_0} a_k$ converges, and if we can determine that $\lim _{n \to \infty } s_n =L$, then $\sum _{k=k_0} a_k=L$.
• If $\lim _{n \to \infty } s_n$ does not exist, then $\sum _{k=k_0} a_k$ diverges.
• If an explicit formula for $s_n$ cannot be found, further analysis is needed.

In the previous section, we studied two types of series where we could find an explicit formula for $s_n$, but unfortunately, this is not always easy or possible. Fortunately, it is not always necessary to do this in order to determine whether $\lim _{n \to \infty } s_n$ exists. Consider the example below.

As it turns out, the above argument can be used to make a very important observation; if $\{a_n\}$ is a sequence for which $\sum _{k=k_0}^{\infty } a_k$ converges, then $\lim _{n \to \infty } a_n =0$. This result is fundamentally important, so we capture it in a theorem.

### The divergence test

Stated in plain English, the above test ensures that if the terms in a sequence do not tend to zero, then we cannot add all of the terms in that sequence together.

This test gives us a quick way to determine if some series diverge.

### Implications of the divergence test

Let’s summarize the important points from the previous discussion.

• If $\sum _{k=k_0}^\infty a_k$ converges, then $\lim _{n \to \infty } a_n =0$.
• If $\lim _{n \to \infty } a_n \neq 0$ (including the case where the limit does not exist), then $\sum _{k=k_0}^{\infty } a_k$ diverges.

While divergence test was straightforward to apply in the previous examples, there is a major point to address about what it does not say.

The divergence test can never be used to conclude that a series converges. The theorem does not state that if $\lim _{n\to \infty } a_n = 0$ then $\sum _{n=1}^\infty a_n$ converges.

We’ve actually seen an example of this in action.

Said another way:

If $\sum _{k=k_0}^{\infty } a_k$ diverges, it’s still possible that $\lim _{n \to \infty } a_n =0$.

To elaborate a little more, we can say that a series $\sum _{k=k_0} a_k$ “passes the divergence test” if $\lim _{n \to \infty } a_n=0$. 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.

There are many questions that require that you now have a firm grasp on the concepts presented thus far. We summarize the important points made thus far, then give many examples that require you to synthesize them.

• There are two fundamental questions we can ask of any sequence.
• Do the numbers in the list approach a finite value?
• Can I sum all of the numbers in the list and obtain a finite result?

These questions can be asked of a given sequence $\{a_n\}$ and can also be asked about $\{s_n\}$ or any sequence constructed from it.

• Given a sequence $\{a_n\}_{n=n_0}$, we construct the sequence of partial sum $\{s_n\}_{n=n_0}$ whose $n$-th term is given by the formula $s_n = \sum _{k=k_0}^n a_k$.
• The symbols $\sum _{k=k_0}^{\infty } a_k$ and $\lim _{n \to \infty } s_n$ are the same.
• By definition $\sum _{k=k_0} a_k$ converges if $\lim _{n \to \infty } s_n$ exists and in this case, the value of each is the same.
• By definition $\sum _{k=k_0} a_k$ diverges if $\lim _{n \to \infty } s_n$ does not exist (which includes if the limit is infinte).
• If the limit of a sequence is not zero, the sum of its terms diverges.
• If a series converges, the limit of the sequence whose terms is being summed is zero.
• If the limit of a sequence is zero, more information is needed to determine whether the sum of its terms converges or diverges.

To answer the following questions, make sure that you understand exactly what is given in the statement of the question first, then try to synthesize the material above.

Suppose $\{a_n\}_{n \geq 1}$ is a sequence and $\sum ^{\infty }_{n= 1} a_n=5$. Let $s_n = \sum ^n_{k=1} a_k$. Select all statements that must be true:
$\lim _{n \to \infty } a_n = 5$ $\lim _{n \to \infty } a_n = 0$ $\lim _{n \to \infty } s_n = 0$ $\lim _{n \to \infty } s_n = 5$ $\sum ^{\infty }_{k=1} s_k$ must diverge. $\sum ^{\infty }_{k=1} (a_k+1) = 5+1=6$ The divergence test tells us $\sum ^{\infty }_{n= 1} a_n$ converges to $L$.