Training Sequences & Series Power Series & Taylor Series
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Power Series & Taylor Series

26 min Sequences & Series

Power Series & Taylor Series

Power Series

A series of the form:

$$\sum_{n=0}^{\infty} c_n (x - a)^n$$

centered at $x = a$ with coefficients $c_n$.

Ratio Test for Convergence

$$L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|$$

If $L < 1$: converges. If $L > 1$: diverges. If $L = 1$: inconclusive.

Taylor Series

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$

When $a = 0$, this is a Maclaurin series.

Common Maclaurin Series

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \qquad \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$

$$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \qquad \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n, \; |x| < 1$$

Example 1

Use the ratio test on $\sum \frac{n!}{3^n}$.

$$L = \lim \frac{(n+1)!/3^{n+1}}{n!/3^n} = \lim \frac{n+1}{3} = \infty$$

Since $L > 1$, the series diverges.

Example 2

Write the first 4 nonzero terms of the Maclaurin series for $e^x$.

$$e^x \approx 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$$

Example 3

Find the radius of convergence: $\sum \frac{x^n}{n}$.

Ratio test: $L = \lim \frac{|x|^{n+1}/(n+1)}{|x|^n/n} = |x| \cdot \lim \frac{n}{n+1} = |x|$.

Converges when $|x| < 1$, so radius $R = 1$.

Practice Problems

1. Use the ratio test on $\sum \frac{1}{n!}$.
2. Find the Maclaurin series for $\sin x$ up to $x^5$.
3. Find the Maclaurin series for $\cos x$ up to $x^4$.
4. Approximate $e^{0.1}$ using 3 terms of the Maclaurin series.
5. What is the radius of convergence of $\sum \frac{x^n}{2^n}$?
6. Use the ratio test on $\sum \frac{2^n}{n!}$.
7. What is a Taylor series centered at $a = 3$?
8. Write $\frac{1}{1-x}$ as a power series and state the interval of convergence.
9. Use the ratio test on $\sum \frac{n}{5^n}$.
10. What is the 4th-degree Taylor polynomial for $\ln(1+x)$ at $a=0$?
11. If the ratio test gives $L = 1$, what do you conclude?
12. Approximate $\sin(0.1)$ using 2 terms of the Maclaurin series.
Show Answer Key

1. $L = \lim \frac{1}{n+1} = 0 < 1$; converges

2. $x - \frac{x^3}{6} + \frac{x^5}{120}$

3. $1 - \frac{x^2}{2} + \frac{x^4}{24}$

4. $1 + 0.1 + 0.005 = 1.105$ (actual ≈ 1.10517)

5. $R = 2$ (ratio test: $|x/2| < 1$)

6. $L = \lim \frac{2}{n+1} = 0$; converges

7. $\sum \frac{f^{(n)}(3)}{n!}(x-3)^n$

8. $\sum_{n=0}^{\infty} x^n$, $|x| < 1$

9. $L = \lim \frac{(n+1)/5^{n+1}}{n/5^n} = 1/5$; converges

10. $x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$

11. Inconclusive — need another test

12. $0.1 - \frac{(0.1)^3}{6} \approx 0.1 - 0.000167 = 0.09983$

Infinite Series & Convergence Placement Test Practice — Sequences & Series