Unit 10 Summary (College Board AP® Calculus BC): Revision Note

Infinite sequences and series summary

Key definitions

• A sequence is an ordered collection of numbers denoted $\left\{a_n\right\}$

• A partial sum $s_n$ is the sum of the first $n$ terms of a sequence

• A series $\sum _{n=0}^{\infty }{a}_n$ is the sum of terms in a sequence

• An infinite series $\sum _{n=0}^n{a}_n$converges if $\underset{n\to \infty }{\lim }{s}_n=\underset{n\to \infty }{\lim }\sum _{k=0}^n{a}_k$ exists

• An infinite series diverges if it does not converge

• A geometric series is of the form $\sum _{n=0}^{\infty }a·r^n=a+ar+a{r}^2+...$

  • $a\ne 0$ is the first term

  • $r$ is the common ratio

• The harmonic series is $\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+...$

• The alternating harmonic series is $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-...$

• A p-series is $\sum _{n=1}^{\infty }\frac{1}{n^p}=1+\frac{1}{2^p}+\frac{1}{3^p}+...$

• A series $\sum _{n=0}^{\infty }{a}_n$ converges absolutely if $\sum _{n=0}^{\infty }\left|a_n\right|$ converges

• A series $\sum _{n=0}^{\infty }{a}_n$ converges conditionally if $\sum _{n=0}^{\infty }{a}_n$ converges but $\sum _{n=0}^{\infty }\left|a_n\right|$ diverges

• A Taylor polynomial of degree $n$ of a function $f$ about the point $x=a$ is $p_n\left(x\right)=f\left(a\right)+f\text{'}\left(a\right)·\left(x-a\right)+f\text{'}\text{'}\left(a\right)·\frac{{\left(x-a\right)}^2}{2!}+...+f^{\left(n\right)}\left(a\right)·\frac{{\left(x-a\right)}^n}{n!}$

  • Provided the $n$ derivatives of $f$ exist at $x=a$

• A Maclaurin polynomial of degree $n$ of a function $f$ is $p_n\left(x\right)=f\left(0\right)+f\text{'}\left(0\right)·x+f\text{'}\text{'}\left(0\right)·\frac{x^2}{2!}+...+f^{\left(n\right)}\left(0\right)·\frac{x^n}{n!}$

  • Provided the $n$ derivatives of $f$ exist at $x=0$

• A power series is a series that depends on a variable $\sum _{n=0}^{\infty }{a}_n·{\left(x-r\right)}^n=a_0+a_1·\left(x-r\right)+a_2·{\left(x-r\right)}^2+...$

• The interval of convergence of a power series is the set of values for which the power series converges

• The radius of convergence is half the length of the interval of convergence

• A Taylor series for a function $f$ about point $x=a$ is $\sum _{n=0}^{\infty }\frac{f^{\left(n\right)}\left(a\right)}{n!}·{\left(x-a\right)}^n=f\left(a\right)+f\text{'}\left(a\right)·\left(x-a\right)+\frac{f\text{'}\text{'}\left(a\right)}{2!}·{\left(x-a\right)}^2+...$

• A Maclaurin series for a function $f$ is $\sum _{n=0}^{\infty }\frac{f^{\left(n\right)}\left(0\right)}{n!}·x^n=f\left(0\right)+f\text{'}\left(0\right)·x+\frac{f\text{'}\text{'}\left(0\right)}{2!}·x^2+...$

Key tests

• nth term test

  • If $\underset{n\to \infty }{\lim }{a}_n\ne 0$, then $\sum _{n=0}^{\infty }{a}_n$ diverges

• Integral test

  • Given that $f$ is a continuous, positive, decreasing function on $[c,\infty )$ such that $a_n=f\left(n\right)$

  • If ${\int }_c^{\infty }f\left(x\right)dx$ exists, then $\sum _{n=c}^{\infty }{a}_n$ converges

  • If ${\int }_c^{\infty }f\left(x\right)dx$ does not exist, then $\sum _{n=c}^{\infty }{a}_n$ diverges

• Comparison test

  • Given that $\sum _{n=1}^{\infty }{a}_n$ and $\sum _{n=1}^{\infty }{b}_n$ are two series with non-negative terms

  • If $\sum _{n=1}^{\infty }{b}_n$ converges and if $a_n\le b_n$ for all $n$, then $\sum _{n=1}^{\infty }{a}_n$ converges

  • If $\sum _{n=1}^{\infty }{b}_n$ diverges and if $a_n\ge b_n$ for all $n$, then $\sum _{n=1}^{\infty }{a}_n$ diverges

• Limit comparison test

  • Given that $\sum _{n=1}^{\infty }{a}_n$ and $\sum _{n=1}^{\infty }{b}_n$ are two series with non-negative terms

  • If $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=L$, where $0<L<\infty$, then either both series converge or both series diverge

• Ratio test

  • $\sum _{n=1}^{\infty }{a}_n$

  • If $\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|<1$, then $\sum _{n=1}^{\infty }{a}_n$ converges absolutely

  • If $\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|>1$ or if the limit is infinite, then $\sum _{n=1}^{\infty }{a}_n$ diverges

  • If $\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|=1$, then the ratio test provides no information about convergence

• Alternating series test

  • Given an alternating series of the form $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}·a_n$ or $\sum _{n=1}^{\infty }{\left(-1\right)}^n·a_n$ with $a_n>0$ for each value of $n$

  • The series converges if $a_1\ge a_2\ge a_3\ge ...\ge a_n\ge ...$ and $\underset{n\to \infty }{\lim }{a}_n=0$

Key formulas

• If $\sum _{n=0}^{\infty }{a}_n=A$ and $\sum _{n=0}^{\infty }{b}_n=B$, where $A$ and $B$ are finite, then:

  • $\sum _{n=0}^{\infty }\left(c{a}_n\right)=cA$ for any real number $c$

  • $\sum _{n=0}^{\infty }\left(a_n+b_n\right)=A+B$

• $\sum _{n=0}^{\infty }a·r^n=\frac{a}{1-r}$ provided $\left|r\right|<1$

• The alternating series error bound for $S=\sum _{n=1}^{\infty }{\left(-1\right)}^n·a_n$ or $S=\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}·a_n$ is given by $\left|S-s_n\right|\le a_{n+1}$ where $s_n=\sum _{k=1}^n{\left(-1\right)}^k·a_k$ or $s_n=\sum _{k=1}^n{\left(-1\right)}^{k+1}·a_k$

• The Lagrange error bound of a Taylor polynomial of degree $p_n$ for a function $f$ about $x=a$ is given by $\left|f\left(x\right)-p_n\left(x\right)\right|\le \frac{M}{\left(n+1\right)!}{\left|x-a\right|}^{n+1}$

  • Where $\left|f^{\left(n+1\right)}\left(x\right)\right|\le M$ for all $x$ in the interval

Key facts

• The following table gives the convergent conditions for common series

• If a power series converges, then one of the following is true:

  • it converges only at a single point

  • it converges for all values in an interval

  • it converges for all values

• The Maclaurin series or a Taylor series of a function can be used to approximate the function at values within its interval of convergence

• The table below shows common Maclaurin series with their interval of convergence

• A power series can be differentiated or integrated term-by-term within its interval of convergence