set_plot_option([svg_file, "maxplot.svg"])$
An infinite series is a sume of infinitely many terms and is written in the form $$\sum_{n=1}^{\infty}a_n = a_1 +a_2 +\cdots$$ and the parital sum of an infinite series is $$S_k = \sum_{n=1}^k a_n = a_1+a_2+\cdots +a_k$$
The partial sum form a sequence $\{S_k\}$. If the sequence of partial sums converges to a real number $S$, the infinite series converges.
If we can describe the convergence of a series to $S$, we call $S$ the sum of the series, and we write
$$\sum_{n=1}^{\infty}a_n=S$$If the sequence of partial sums diverges, we have the divergence of a series.
The harmonic series is defined as $$\sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac13+\cdots$$
$\newcommand{\SUM}[2]{\sum_{n=1}^{#1}\left(#2\right)}$ Let $\SUM{\infty}{a_n}$ and $\SUM{\infty}{b_n}$ be convegent series. Then the following algebraic properties hold.
A geometric series is a series of the form
$$\SUM{\infty}{ar^{n-1}}=a+ar+ar^2+ar^3+\cdots$$were $|r|<1$.
A telescoping series is a series in which most of the terms cancel in each of the partial sums, leaving only some of the first terms and some of the last terms. For example: $$\SUM{k}{b_n-b_{n+1}} = b_1-b_{k+1}$$
sum(1/2,i,1,10);
10*(1/2);
(1/2)^1+(1/2)^2+(1/2)^3+(1/2)^4;
sum((1/2)^n,n,1,4);
plot2d([discrete,makelist((1/2)^i,i,4)],[style,points])$
As you can see above instead of evaluating:
$$(1/2)^1+(1/2)^2+(1/2)^3+(1/2)^4 = 15/16$$we compact the notation to: $$\sum_{n=1}^4 (1/2)^n = 15/16$$
If $\lim_{n\to \infty}a_n=c\ne 0$ or $\lim_{n\to \infty}a_n$ does not exist, then the series $\SUM{\infty}{a_n}$ diverges.
Consider the series $\SUM{\infty}{a_n}$.
Then $$\SUM{\infty}{a_n} \text{ and } \int_N^{\infty}f(x)\,dx$$ both converge or both diverge.
Very important to satisfy the three conditions in order to use the Integral test.
For any real number $p$, the series $$\SUM{\infty}{\frac{1}{n^p}}$$ is called a $p$-series.
(More will follow after comparison tests like the ratio test and root test) Given the $p$-series $\SUM{\infty}{\frac{1}{n^p}}$
Suppose $\SUM{\infty}{a_n}$ is a convergent series with positive terms. Suppose there exists a function $f$ satisfying the following three conditions:
Let $S_N$ be the $N^{\text{th}}$ partial sum of $\SUM{\infty}{a_n}$. For all positive integers $N$,
$$S_N+\int_{N+1}^{\infty}f(x)\,dx<\SUM{\infty}{a_n}<S_N+\int_N^{\infty}f(x)\,dx$$In other words, the remainder $$R_N=\SUM{\infty}{a_n}-S_N = \sum_{n=N+1}^{\infty}\left(a_n\right)$$ satisfies the following estimate:
$$\int_{N+1}^{\infty}f(x)\,dx < R_n < \int_N^{\infty}f(x)\,dx.$$Consider $\SUM{\infty}{\frac{1}{n^3}}$
sum(1/n^3,n,1,10);
float(sum(1/n^3,n,1,10));
f(x):=1/x^3;
integrate(f(x),x);
assume(N>1)$
integrate(f(x),x,1,N);
limit(integrate(f(x),x,1,N),N,infinity);
As we can see from the calculations the remainder estimate is $R_n=\frac{1}{2N^2}$. So if we want the error for $S_{10}$ we have:
Rn(N):=1/(2*N^2);
Rn(10);
float(Rn(10));
Now determine least value of $N$. That is solve the following inequality: $$R_N<0.001 \text{ or } \frac{1}{2N^2}<0.001$$
assume(N>1)$
solve(1/(2*N^2)=0.001,N);
float(solve(1/(2*N^2)=0.001,N));
Therefore, the minimum necessary value is $N=23$.
Let $a_n$, $b_n\ge 0$ for all $n\ge 1$.
Any series whose terms alternate between positive and negative values is called an alternating series. An alternating series can be written in the form
$$\SUM{\infty}{(-1)^{n+1}b_n} = b_1 -b_2+b_3-b_4+\cdots$$or $$\SUM{\infty}{(-1)^n b_n} = -b_1+b_2-b_3+b_4-\cdots$$
where $b_n\ge 0$ for all positive integers $n$.
An alternating series of the form $$\SUM{\infty}{(-1)^{n+1}b_n}\text{ or } \SUM{\infty}{(-1)^n b_n}$$ converges if
Consider the alternating series of the form $$\SUM{\infty}{(-1)^{n+1}b_n} \text{ or }\SUM{\infty}{(-1)^nb_n}$$ that satisfies the hypotheses of the alternating series test.
Let $S$ denote the sum of the series and $S_N$ denote the $N^{\text{th}}$ partial sum. For any integer $N\ge 1$, the remainder $R_N = S - S_N$ satisfies $$|R_N|\le b_{N+1}$$
A series $\SUM{\infty}{a_n}$ exhibits absolute convergence if $\SUM{\infty}{|a_n|}$ converges. A series $\SUM{\infty}{a_n}$ exhibits conditional convergence if $\SUM{\infty}{a_n}$ converges but $\SUM{\infty}{|a_n|}$ diverges.
If $\SUM{\infty}{|a_n|}$ converges, then $\SUM{\infty}{a_n}$ converges.
Let $\SUM{\infty}{a_n}$ be a series with nonzero terms. Let $$\rho = \lim_{n\to \infty}\left|\frac{a_{n+1}}{a_n}\right|.$$
Consider the series $\SUM{\infty}{a_n}$. Let $$\rho = \lim_{n\to \infty}\sqrt[n]{\left|a_n\right|}$$
| Series or Test | Conclusions | Comments |
|---|---|---|
|
Divergence Test For any series $\sum_{n=1}^{\infty}(a_{n})$, evaluate $\lim_{n\to\infty}(a_{n})$. |
If $\lim_{n\to\infty}(a_{n})=0$, the test is inconclusive. | The test cannot prove convergence of a series. |
| If $\lim_{n\to\infty}(a_{n})\ne0$, the series diverges. | ||
|
Geometric Series $\sum_{n=1}^{\infty}\left(ar^{n-1}\right)$ |
If $|r|<1$, the series converges to $\frac{a}{1-r}$. | Any geometric series can be re-indexed to be written in the form $a+ar+ar^{2}+\cdots$, where $a$ is the initial term and $r$ is the ratio. |
| If $|r|\ge1$, the series diverges. | ||
|
$p$-Series $\sum_{n=1}^{\infty}\left(\frac{1}{n^{p}}\right)$ |
If $p>1$, the series converges. | For $p=1$, we have the harmonic series $\sum_{n=1}^{\infty}\left(\frac{1}{n}\right)$. |
| If $p\le1$, the series diverges. | ||
|
Comparison Test For $\sum_{n=1}^{\infty}(a_{n})$ with nonnegative terms, compare with a known series $\sum_{n=1}^{\infty}(b_{n})$. |
If $a_{n}\le b_{n}$ for all $n\ge N$ and $\sum_{n=1}^{\infty}(b_{n})$ converges, then $\sum_{n=1}^{\infty}(a_{n})$ converges | Typically used for a series similar to a geometric or $p$-series. It can sometimes be difficult to find an appropriate series. |
| If $a_{n}\ge b_{n}$ for all $n\ge N$ and $\sum_{n=1}^{\infty}(b_{n})$ diverges, then $\sum_{n=1}^{\infty}(a_{n})$ diverges. | ||
|
Limit Comparison Test For $\sum_{n=1}^{\infty}(a_{n})$ with positive terms, compare with a series $\sum_{n=1}^{\infty}(b_{n})$ by evaluating $L=\lim_{n\to\infty}\frac{a_{n}}{b_{n}}$. |
If $L$ is a real number and $L\ne0$, then $\sum_{n=1}^{\infty}(a_{n})$ and $\sum_{n=1}^{\infty}(b_{n})$ both converge or both diverge. | Typically used for a series similar to a geometric or p-series. Often easier to apply than the comparison test. |
| If $L=0$ and $\sum_{n=1}^{\infty}(b_{n})$ converges, then $\sum_{n=1}^{\infty}(a_{n})$ converges. | ||
| If $L=\infty$ and $\sum_{n=1}^{\infty}(b_{n})$ diverges, then $\sum_{n=1}^{\infty}(a_{n})$ diverges. | ||
Integral Test If there exists a positive, continuous, decreasing function $f$ such that $a_{n}=f(n)$ for all $n\ge N,$evaluate $\int_{N}^{\infty}f(x)\,dx$. |
$\int_{N}^{\infty}f(x)\,dx$ and $\sum_{n=1}^{\infty}(a_{n})$ both converge or both diverge. | Limited to those series for which the corresponding function $f$ can be easily integrated. |
Alternating Series $\sum_{n=1}^{\infty}\left((-1)^{n+1}b_{n}\right)$ or $\sum_{n=1}^{\infty}\left((-1)^{n}b_{n}\right)$ |
If $b_{n+1}\le b_{n}$ for all $n\ge1$ and $b_{n}\to0$ as $n\to\infty$, then the series converges. | Only applies to alternating series. |
|
Ratio Test For any series $\sum_{n=1}^{\infty}(a_{n})$ with nonzero terms, let $\rho=\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_{n}}\right|$. |
If $0\le\rho<1$, the series converges absolutely. | Often used for series involving factorials or exponential. |
| If $\rho>1$ or $\rho=\infty$, the series diverges. | ||
| If $\rho=1$, the test is inconclusive. | ||
|
Root Test For any series $\sum_{n=1}^{\infty}(a_{n})$ with nonzero terms, let $\rho=\lim_{n\to\infty}\sqrt[n]{|a_{n}|}$. |
If $0\le\rho<1$, the series converges absolutely. | Often used for series where $|a_n|=b_n^n$ |
| If $\rho>1$ or $\rho=\infty$, the series diverges. | ||
| If $\rho=1$, the test is inconclusive. |