Notes
Series basics
A series $\displaystyle \sum a_n$ converges if its partial sums $S_n = a_1 + \dots + a_n$ approach a finite limit.
- Geometric series $\displaystyle \sum_{n=0}^{\infty} ar^n$ converges to $\displaystyle \frac{a}{1 - r}$ when $|r| < 1$ and diverges otherwise.
- nth term test: if $\displaystyle \lim a_n \ne 0$, the series diverges. (If the limit is 0, the test tells you nothing.)
- p-series $\displaystyle \sum \frac{1}{n^p}$ converges when $p > 1$. The harmonic series ($p = 1$) diverges.
Convergence tests
| Test | Use when | Conclusion | ||
|---|---|---|---|---|
| Integral | $a_n = f(n)$, $f$ positive, decreasing | Series and $\displaystyle \int_1^\infty f$ do the same thing | ||
| Comparison / limit comparison | Looks like a known series | Same behavior as the known series | ||
| Alternating series | Signs alternate, $ | a_n | $ decreases to 0 | Converges |
| Ratio | Factorials or $n$th powers | $L < 1$ converges, $L > 1$ diverges, $L = 1$ no info |
- Absolute convergence: $\displaystyle \sum |a_n|$ converges. Conditional: $\displaystyle \sum a_n$ converges but $\displaystyle \sum |a_n|$ doesn't, like $\displaystyle \sum \frac{(-1)^n}{n}$.
- Alternating series error bound: the error after stopping is at most the size of the first omitted term.
Taylor and Maclaurin series
The Taylor series of $f$ about $x = a$:
| Function | Maclaurin series | Converges for | ||
|---|---|---|---|---|
| $e^x$ | $\displaystyle 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ | all $x$ | ||
| $\sin x$ | $\displaystyle x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$ | all $x$ | ||
| $\cos x$ | $\displaystyle 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$ | all $x$ | ||
| $\displaystyle \frac{1}{1 - x}$ | $1 + x + x^2 + x^3 + \dots$ | $ | x | < 1$ |
- Build new series by substituting, multiplying, differentiating or integrating known ones. $\displaystyle e^{-x^2} = 1 - x^2 + \frac{x^4}{2!} - \dots$
- Lagrange error bound: $\displaystyle |R_n(x)| \le \frac{M}{(n+1)!}|x - a|^{n+1}$, where $M$ bounds $|f^{(n+1)}|$ between $a$ and $x$.
Power series and the interval of convergence
Use the ratio test to find the radius, then test both endpoints separately.
Topics in this unit
Each topic has its own short notes and practice.
- 10.1Defining Convergent and Divergent Infinite Series
- 10.2Working with Geometric Series
- 10.3The nth Term Test for Divergence
- 10.4Integral Test for Convergence
- 10.5Harmonic Series and p-Series
- 10.6Comparison Tests for Convergence
- 10.7Alternating Series Test for Convergence
- 10.8Ratio Test for Convergence
- 10.9Determining Absolute or Conditional Convergence
- 10.10Alternating Series Error Bound
- 10.11Finding Taylor Polynomial Approximations of Functions
- 10.12Lagrange Error Bound
- 10.13Radius and Interval of Convergence of Power Series
- 10.14Finding Taylor or Maclaurin Series for a Function
- 10.15Representing Functions as Power Series
Practice
Geometric with $a = 6$ and $\displaystyle r = \frac{1}{3}$: $\displaystyle \frac{6}{1 - 1/3} = 9$.
A p-series with $p = 2 > 1$ converges. The last one fails the nth term test.
It converges by the alternating series test, but $\displaystyle \sum \frac{1}{n}$ diverges.
$\displaystyle \frac{a_{n+1}}{a_n} = \frac{3}{n + 1} \to 0$, so the series converges.
$\displaystyle \frac{(2x)^3}{3!} = \frac{8x^3}{6}$.
Substitute $x^2$ into $\displaystyle x - \frac{x^3}{3!}$.
It's geometric with ratio $\displaystyle \frac{x - 2}{3}$; converges when $|x - 2| < 3$.
The error is at most the first omitted term, $\displaystyle \frac{1}{4^2}$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Let $f(x) = \sum_{n=1}^{\infty} \frac{(x - 1)^n}{n \cdot 2^n}$.
(a) Find the radius of convergence.
Show the answer and scoring guide
Ratio test: $\displaystyle \left|\frac{(x - 1)^{n+1}}{(n + 1)2^{n+1}} \cdot \frac{n \cdot 2^n}{(x - 1)^n}\right| \to \frac{|x - 1|}{2}$. Converges when $|x - 1| < 2$, so the radius is 2.
Check off the points you earned:
(b) Find the interval of convergence. Justify the endpoints.
Show the answer and scoring guide
At $x = 3$: $\displaystyle \sum \frac{1}{n}$, the harmonic series, diverges. At $x = -1$: $\displaystyle \sum \frac{(-1)^n}{n}$ converges by the alternating series test. Interval: $-1 \le x < 3$.
Check off the points you earned:
(c) Write the first three nonzero terms of the series for $f'(x)$.
Show the answer and scoring guide
$\displaystyle f'(x) = \sum \frac{(x - 1)^{n-1}}{2^n} = \frac{1}{2} + \frac{x - 1}{4} + \frac{(x - 1)^2}{8} + \dots$
Check off the points you earned: