Series - Tests for Convergence/Divergence

Important Series

There are two series that are important to know for a variety of reasons. In particular, they are useful for comparison tests.

Geometric series. These are series with a common ratio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} between adjacent terms which are usually written

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^{\infty}a_{0}r^{k}.}

These are convergent if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |r|<1} , and divergent if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |r|\geq1} . If it is convergent, we can find the sum by the formula

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\frac{a_{0}}{1-r},}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{0}} is the first term in the series (if the index starts at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=2} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=6} , then "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{0}} " is actually the first term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{2}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{6}} , respectively).

p-series. These are series of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^{\infty}\frac{1}{k^{p}}.}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p>1} , then the series is convergent. On the other hand, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\leq1} , the p-series is divergent.

The Divergence Test

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \lim_{k\rightarrow\infty}a_{k}\neq0,}}

then the series/sum

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^{\infty}a_{k}}

diverges.

Note: The opposite result doesn't allow you to conclude a series converges. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \lim_{k\rightarrow\infty}a_{k}=0}}  , it merely indicates the series might converge, and you still need to confirm it through another test.

The Integral Test

Suppose the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is continuous, positive and decreasing on some interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [b,\infty)} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\geq1} , and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{k}=f(k)} . Then the series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=b}^{\infty}a_{k}} is convergent if and only if for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\geq b} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{c}^{\infty}f(x)\, dx}

is convergent (not infinite).

Note: This test, like many of them has a few requirements. In order to use it on a test, you need to state/show:

• For all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\geq c} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\geq b} , the function is positive. (Most of the time, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is just my starting index Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} ).
• For all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\geq c} , the function is decreasing.
• The integral is convergent (or divergent, if you're proving divergence).

Then, you can say, "By the Integral Test, the series is convergent (or divergent)."

I wrote this with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} instead of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} for a lower bound to indicate you only need to show the series and function are "eventually" decreasing, positive, etc. In other words, we don't care what happens at the beginning (or head) of a series - only at the end (or tail).

The Comparison Test

Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^{\infty} a_{k}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^{\infty} b_{k}} are series with positive terms. Then

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{k}\leq b_{k}} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^{\infty} b_{k}} is convergent, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^{\infty} a_{k}} is convergent.

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{k}\geq b_{k}} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^{\infty} b_{k}} is divergent, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^{\infty} a_{k}} is divergent.

Note: Requirements for this test include showing (or at least stating): \begin{itemize} \item For all $k\geq c$ for some $c$ bigger than our starting index, $a_{k}$ is positive. (Most of the time, $c$ is just the starting index.) \item For all $k\geq c$, $a_{k}\leq b_{k}$ for convergence, or $a_{k}\geq b_{k}$ for divergence. \item \textbf{This is important - }State why $\sum b_{k}$ is convergent, such as a $p$-series with $p>1$, or a geometric series with $|r|<1.$ Obviously, you would need to state why it is divergent if you're showing it's divergent. \end{itemize} \textbf{\uline{Then}}, you can say, ``By the Comparison Test, the series is convergent (or divergent).

The Limit Comparison Test

Suppose $\sum a_{k}$ and $\sum b_{k}$ are series with positive terms. If \[ \lim_{k\rightarrow\infty}\frac{a_{k}}{b_{k}}=c \]

where $0<c<\infty,$ then either both series converge, or both series

diverge. Additionally, if $c=0$ and $\sum b_{k}$ converges, $\sum a_{k}$ also converges. Similarly, if $c=\infty$ and $\sum b_{k}$ diverges, then $\sum a_{k}$ also diverges.

\emph{\uline{Notes}}\emph{: }First of all, let's mention the idea here. If some series $\sum b_{k}$ converges, then \[ \sum cb_{k} \]

converges where $c\neq\pm\infty$ is a constant. This test shows

that one series \emph{\uline{eventually}} is just like the other one multiplied by a constant, and for that reason it will also converge/diverge if the one compared converges/diverges. To use it, you need to state/show: \begin{itemize} \item $a_{k}$ is always positive (really, non-negative). \item ${\displaystyle \lim_{k\rightarrow\infty}\frac{a_{k}}{b_{k}}}=c$. \item State why $\sum b_{k}$ is convergent, such as a $p$-series with $p>1$, or a geometric series with $|r|<1.$ Obviously, you would need to state why it is divergent if you're showing it's divergent. \end{itemize} \textbf{\uline{Then}}, you can say, ``By the Limit Comparison Test, the series is convergent (or divergent).

Like the Comparison Test and the integral test, it's fine if the first terms are kind of ``wrong - negative, for example - as long as they eventually wind up (for $k>c$ for a particular $c$) meeting the requirements.

\hrulefill

The Alternating Series Test

If a series $\sum a_{k}$ is \begin{enumerate} \item Alternating in sign, and \item ${\displaystyle \lim_{k\rightarrow0}}|a_{k}|=0,$ \end{enumerate} then the series is convergent.

\emph{\uline{Notes}}\emph{: }This is a fairly straightfoward test. You only need to do two things: \begin{enumerate} \item Mention the series is alternating (even though it's usually obvious). \item Show the limit converges to zero. \end{enumerate} \textbf{\uline{Then}}, you can say, ``By the Alternating Series Test, the series is convergent.

As an additional detail, if it fails to converge to zero, then you would say it diverges by the Divergence Test, \textbf{\uline{NOT}} the Alternating Series Test.

The Ratio Test

Let $\sum a_{k}$ be a series. Then: \begin{enumerate} \item if ${\displaystyle \lim_{k\rightarrow\infty}\left|\frac{a_{k+1}}{a_{k}}\right|=L<1,}$ the series is absolutely convergent (and therefore convergent),\\

\item if ${\displaystyle \lim_{k\rightarrow\infty}\left|\frac{a_{k+1}}{a_{k}}\right|=L>1}$ or ${\displaystyle \lim_{k\rightarrow\infty}\left|\frac{a_{k+1}}{a_{k}}\right|=L=\infty,}$ the series is divergent,\\

\item if ${\displaystyle \lim_{k\rightarrow\infty}\left|\frac{a_{k+1}}{a_{k}}\right|=L=1,}$ the Ratio Test is inconclusive.\\

\end{enumerate} \emph{\uline{Notes}}\emph{: }Both this and the Root Test have the least requirements. The Ratio Test \emph{\uline{does}} require that such a limit exists, so a series like \[ 0+1+0+\frac{1}{4}+0+\frac{1}{9}+\cdots \]

could not be assessed as written with the Ratio Test, as division

by zero is undefined. You might have to argue it's the same sum as \[ 1+\frac{1}{4}+\frac{1}{9}+\cdots, \]

and then you could apply the Ratio Test.

The Root Test

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle\sum_{k=0}^{\infty} a_{k}} be a series. Then:

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \lim_{k\rightarrow\infty}\sqrt[k]{|a_{k}|}}=L<1,} the series is absolutely convergent (and therefore convergent).

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \lim_{k\rightarrow\infty}\sqrt[k]{|a_{k}|}}=L>1} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle\lim_{k\rightarrow\infty}\sqrt[k]{|a_{k}|}}=L=\infty,}

the series is divergent.

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \lim_{k\rightarrow\infty}\sqrt[k]{|a_{k}|}}=L=1} , the Root Test is inconclusive.