Taylor Series

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 11.10 – Taylor and Maclaurin Series
• Section 11.11 – Applications of Taylor Polynomials

Taylor Series

Applications

Investigation 13

Find the sum of the following series:

1. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{4^n}{n{5}^n}}$$\displaystyle \sum_{n\,=\,1}^{\infty}{(-1)^n\frac{4^n}{n5^n}}$
2. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{4^n}{n!5^n}}$$\displaystyle \sum_{n\,=\,1}^{\infty}{\frac{4^n}{n!5^n}}$
3. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{{\pi }^{2n}}{4^{2n}(2n)!}}$$\displaystyle \sum_{n\,=\,0}^{\infty}{(-1)^n\frac{\pi^{2n}}{4^{2n}(2n)!}}$

Investigation 14

Explain why there is no antiderivative to ${\displaystyle \int e^{x^2}dx}$$\displaystyle \int{e^{x^2}\,dx}$

Investigation 15

1. Rewrite ${\displaystyle e^{x^2}}$$\displaystyle e^{x^2}$ as a power series.
2. Find the antiderivative of the power series for ${\displaystyle e^{x^2}}$$\displaystyle e^{x^2}$.
3. Evaluate: ${\displaystyle {\int }_0^1{e}^{x^2}dx}$$\displaystyle \int_0^1{e^{x^2}\,dx}$

Investigation 16

Explain why there is no antiderivative to ${\displaystyle \int \cos \left(x^3\right)dx}$$\displaystyle \int{\cos{\left(x^3\right)}\,dx}$

Investigation 17

1. Rewrite ${\displaystyle \cos \left(x^3\right)}$$\displaystyle \cos{\left(x^3\right)}$ as a power series.
2. Find the antiderivative of the power series for ${\displaystyle \cos \left(x^3\right)}$$\displaystyle \cos{\left(x^3\right)}$.
3. Evaluate: ${\displaystyle \int \cos \left(x^3\right)dx}$$\displaystyle \int{\cos{\left(x^3\right)}\,dx}$.
4. Evaluate: ${\displaystyle {\int }_0^{\pi /6}\cos \left(x^3\right)dx}$$\displaystyle \int_0^{\pi/6}{\cos{\left(x^3\right)}\,dx}$.

Investigation 18

Use any method to integrate the following:

1. ${\displaystyle {\int }_0^{1/2}\frac{1}{1-x^5}dx}$$\displaystyle \int_0^{1/2}{\frac{1}{1-x^5}\,dx}$
2. ${\displaystyle \int \ln (1+x^3)dx}$$\displaystyle \int{\ln{\left(1+x^3\right)}\,dx}$
3. ${\displaystyle \int x^2\sin \left(x^2\right)dx}$$\displaystyle \int{x^2\sin{\left(x^2\right)}\,dx}$
4. ${\displaystyle \int \frac{e^x-1}{x}dx}$$\displaystyle \int{\frac{e^x-1}{x}\,dx}$
5. ${\displaystyle \int {\tan }^{-1}\left(x^2\right)dx}$$\displaystyle \int{\tan^{-1}{\left(x^2\right)}\,dx}$

Investigation 19

Use a series to evaluate the following:

1. ${\displaystyle \underset{x\to 0}{lim}\frac{1-\cos (x)}{1+x-e^x}}$$\displaystyle \lim_{x\rightarrow 0}{\frac{1-\cos{(x)}}{1+x-e^x}}$

Homework

At this time, you should be able to complete the following assignments:

• Section 11.10: # 53, 55.
• Section 11.9: # 1, 3, 5, 7, 9, 11.

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Created: Tuesday, 1 December 2020 6:38 EDT Last Modified: Wednesday, 29 June 2022 - 18:42 (EDT)