Testing Series
Published in Summer, 2024
1. Limit Test
Theorem Let $(a_n)$ be an infinite series, if $\displaystyle{\sum_{n = 1}^{\infty} a_n}$ converges, then $\displaystyle{\lim_{n \rightarrow \infty} a_n= 0}$.
It is important to know that the reverse is not true.
Theorem Let $(a_n)$ be an infinite series, if $\displaystyle{\lim_{n \rightarrow \infty} a_n \neq 0}$ or doesn’t exist, then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.
Exercises:
Justify if the following series converge or diverge.
- \[\sum_{n=1}^{\infty}\frac{n^2}{5n^2 + 4}\]
- \[\sum_{n=1}^{\infty} (\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}})\]
- \[\sum_{n=2}^{\infty} \frac{1}{n^3 - n}\]
2. Integral Test
Theorem Let $(a_n)$ be an infinite series, and $f(x)$ is a continuous function defined on $(0,\infty)$. If for all positive integer $n$, we have $a_n = f(n)$, then by integral test, we say $\displaystyle{\sum_{n = 1}^{\infty} a_n}$ converges if the followings hold:
$f(x) \geq 0$ for all $n \geq 1.$
$f(x)$ is decreasing on $(a,\infty)$ for some $a \geq 1$, and $\lim_{n \rightarrow \infty} f(x) = 0$.
$\displaystyle{\int_1^{\infty} f(x) dx}$ converges.
Exercises : Justify if the following series converge or diverge.
$\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$
$\sum_{n=1}^{\infty} \frac{\ln (n)}{n}$
$\sum_{n=1}^{\infty} ne^{-n^2}$
Find the values of $p$, such that the series $\displaystyle{\sum_{n=1}^{\infty} n(1+n^2)^p}$ is convergent.
3. Comparison Test
Theorem Let $(a_n), (b_n)$ be \textbf{positive}, infinite series with the property that $a_n \leq b_n$, then
If $\displaystyle{\sum_{n=1}^{\infty} b_n}$ is convergent, then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is convergent.
If $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is divergent, then $\displaystyle{\sum_{n=1}^{\infty} b_n}$ is divergent.
Exercises: Justify if the following series converge or diverge.
$\displaystyle{\sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3}}$
$\displaystyle{\sum_{n=1}^{\infty} \frac{\ln (n)}{n}}$
Theorem Let $(a_n), (b_n)$ be positive, infinite series, if $\displaystyle{\lim_{n \rightarrow \infty}\frac{a_n}{b_n}}$ exists, then both $\displaystyle{\sum_{n=1}^{\infty} a_n}$ and $\displaystyle{\sum_{n=1}^{\infty} b_n}$ converges or diverges.
Exercises : Justify if the following series converge or diverge.
$\displaystyle{\sum_{n=1}^{\infty} \frac{1}{2^n - 3}}$
$\displaystyle{\sum_{n=1}^{\infty} \sin(\frac{1}{n^2})}$
4. Alternating Series Test
Theorem Let $(b_n)$ be an infinite series where $b_n \geq 0$ for all $n$, if $b_{n+1} \leq b_n$ and $\lim_{n \rightarrow \infty} b_n = 0$, then $\displaystyle{\sum_{i=1}^{\infty} (-1)^{n} b_n}$ and $\displaystyle{\sum_{i=1}^{\infty} (-1)^{n+1} b_n}$ converge.
Exercises : Justify if the following series converge or diverge.
$\displaystyle{\sum_{n=1}^{\infty}(-1)^n \frac{1}{n}}$
$\displaystyle{\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^2}{n^3+1}}$
Definition : Let $(a_n)$ be an infinite series, the sum $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is called absolutely convergent if $\displaystyle{\sum_{n=1}^{\infty} \vert a_n \vert}$ converges. Series that converge but not absolutely converge are called conditionally convergent.
Exercise : Test the sum $\displaystyle{\sum_{n=1}^{\infty} \frac{\cos(n)}{n^2}}$.
5. Ratio Test and Root Test
Theorem Let $(a_n)$ be an infinite series, define $L = \displaystyle{\lim_{n \rightarrow \infty} \Big| \frac{a_{n+1}}{a_n} \Big|}$, then:
If $L < 1$, then $\displaystyle{\sum_{n = 1}^{\infty} a_n}$ is absolutely convergent.
If $L > 1$, then $\displaystyle{\sum_{n = 1}^{\infty} a_n}$ is divergent.
If $L=1$, then the ratio test is inconclusive
Exercises : Justify if the following series converge or diverge.
$\displaystyle{\sum_{n=1}^{\infty}\frac{(-10)^n}{4^{2n+1}(n+1)}}$
$\displaystyle{\sum_{n=1}^{\infty}\frac{n^n}{n!}}$
Theorem Let $(a_n)$ be an infinite series, define $L = \displaystyle{\lim_{n \rightarrow \infty} \sqrt[n]{\vert a_n \vert}}$, then:
If $L < 1$, then $\displaystyle{\sum_{n = 1}^{\infty} a_n}$ is absolutely convergent.
If $L > 1$, then $\displaystyle{\sum_{n = 1}^{\infty} a_n}$ is divergent.
If $L=1$, then the root test is inconclusive \end{theorem}
Exercises : Justify if the following series converge or diverge.
- $\displaystyle{\sum_{n=1}^{\infty}\frac{n^{n+1}}{3^{2n+1}}}$\
More series tests like Dirichlet’s test, Abel’s test, Weierstrass M-test, are beyond the slope of this course and hence will not be introduced. You can look it up if you are interested.
6. Strategy for Testing Series
If $(a_n)$ is of the form $\displaystyle{ \frac{1}{n^p}}$, then, we should know that $\displaystyle{\sum a_n}$ converges for $p>1$ and diverges for $p \leq 1$, also if $(a_n)$ is a geometric series, then we can use the formula to find its sum.
If $(a_n)$ has a form which is similar to $p$-series or geometric series, then we can try to use comparison tes
If $\lim_{n \rightarrow \infty} a_n \neq 0$, then $(a_n)$ is divergent.
If $(a_n) = (-1)^n b_n$, then we shall apply alternating series test.
If inside $(a_n)$ there are terms like $n!$, or some products related to $n$, then we shall consider ratio test.
If $a_n = (b_n)^n$. then we shall consider root test.
If $a_n = f(n)$ for some positive elementary function $f(x)$, if it is easy to integrate $f(x)$, we shall consider integral test.
Exercises : Justify if the following series converge or diverge.
$\displaystyle{\sum_{n=1}^{\infty}\frac{\sqrt{n^3+1}}{3n^3 + 4n^2 + 2}}$
$\displaystyle{\sum_{n=0}^{\infty} (-1)^n\frac{\pi^{2n}}{(2n)!}}$
$\displaystyle{\sum_{n=1}^{\infty} \frac{n \ln (n)}{(1+n)^3}}$
$\displaystyle{\sum_{n=1}^{\infty} \frac{\sin(2n)} {1+2^n}}$
$\displaystyle{\sum_{n=1}^{\infty} \frac{e^{1/n}}{n^2}}$
$\displaystyle{\sum_{n=1}^{\infty} \frac{1}{n + n \cos^n (n) }}$
$\displaystyle{\sum_{n=1}^{\infty} \frac{1} {\tan(n)}}$
- $\displaystyle{\sum_{n=1}^{\infty} \frac{5^n}{3^n+4^n}}$
- $\displaystyle{\sum_{n=1}^{\infty} \frac{1}{(n+1)\sqrt{n^2 + 1}}}$
7. Power Series, Radius of Convergence and Interval of Convergence
Theorem For any given power series $a_n = \displaystyle{\sum_{n=0}^{\infty} c_n(x-a)^n}$, there are only three possibilities:
The series converges only when $x=a$;
The series converges for all $x$;
There exists a positive number $R$ such that the series converges if $\vert x - a \vert < R$ and diverges if $\vert x - a \vert > R$. In this case, $R$ is called the \textbf{radius of convergence}. The interval of convergence is one of $(a-R,a+R);(a-R,a+R];[a-R,a+R);[a-R,a+R]$, depending on the convergence of boundary points.
Exercises :
For what values of $x$ is the series $\displaystyle{\sum_{n=0}^{\infty} n!x^n}$ convergent?
For what values of $x$ is the series $\displaystyle{J_0(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{2^{2n}(n!)^2}}$ convergent?
Find the radius of convergence and the interval of convergence of the series $\displaystyle{\sum_{n=1}^{\infty} (-1)^n\frac{x^n}{\sqrt[3]{n}}}$
Recommended citation: Jiajun Zhang, (2024). Testing Series