# Section 6.2 Calculus 2

Taylor and Maclaurin Series

## 6.2 Taylor and Maclaurin Series

### 6.2.1 Power Series from Functions

• Let $$f(x)$$ have derivatives of all orders nearby $$a$$. Then the Taylor series generated by $$f$$ at $$a$$ is given by $$\sum_{k=0}^\infty\frac{f^{(k)}(a)}{k!}(x-a)^k$$, where $$f^{(k)}(a)$$ is the $$k^{th}$$ derivative of $$f$$ at $$a$$.
• A Maclaurin series is a Taylor series where $$a=0$$.
• A Taylor/Maclaurin series is said to converge to its generating function if it is equal to it for all members of its domain.
• Example Let $$f(x)=\frac{1}{1+x}$$ with the domain $$-1<x<1$$, and guess a formula for $$f^{(k)}(0)$$ by computing its first few terms. Then show that the Maclaurin series generated by $$f$$ converges to $$f$$.
• Example Let $$g(x)=\frac{2}{x}$$ with the domain $$0<x<4$$, and guess a formula for $$g^{(k)}(2)$$ by computing its first few terms. Then show that the Taylor series generated by $$g$$ at $$2$$ converges to $$g$$.
• It can be shown that $$f$$ defined by $$f(0)=0$$ and $$f(x)=e^{-1/x^2}$$ otherwise satisfies $$f^{(k)}(0)=0$$, giving an example of a function which doesn’t converge to its Taylor series.
• If a power series of the form $$\sum_{k=0}^\infty\frac{f^{(k)}(a)}{k!}(x-a)^k$$ converges to $$f(x)$$, then that power series is the Taylor series generated by $$f(x)$$ at $$a$$.

### 6.2.2 Maclaurin Series for $$e^x$$, $$\sin x$$, $$\cos x$$

• The following Maclaurin Series can be shown to converge to their generating functions:
• $$e^x=\sum_{k=0}^\infty\frac{x^k}{k!} = 1+x+\frac{x^2}{2}+\frac{x^3}{6}+\dots$$
• $$\cos x = \sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!} = 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots$$
• $$\sin x = \sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!} = x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+\dots$$
• Example Show how to generate the Maclaurin series for $$e^x$$.
• Example Show how to generate the Maclaurin series for $$\sin x$$.

### Review Exercises

1. Let $$f(x)=\frac{1}{1-x}$$ with the domain $$-1<x<1$$, and guess a formula for $$f^{(k)}(0)$$ by computing its first few terms. Then show that the Maclaurin series generated by $$f$$ converges to $$f$$.
2. Let $$g(x)=\frac{3}{x}$$ with the domain $$0<x<6$$, and guess a formula for $$g^{(k)}(3)$$ by computing its first few terms. Then show that the Taylor series generated by $$g$$ at $$3$$ converges to $$g$$.
3. Let $$h(x)=\frac{1}{x^2+1}$$ with the domain $$-1<x<1$$. It may be shown that the first few terms of $$\<h^{(k)}(0)\>_{k=0}^\infty$$ are $$\<1,0,-2,0,24,0,-720,\dots\>$$. Show that the Maclaurin series generated by $$h$$ converges to $$h$$.
4. Show how to generate the Maclaurin series $$\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}$$ for $$\cos x$$.
5. Find the Maclaurin series for $$\sinh x$$.
6. Find the Maclaurin series for $$\cosh x$$.
7. Find the Maclaurin series for $$e^{-x}$$.
8. Find the Maclaurin series for $$x^3+3x-7$$.

Solutions

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 9.8