5 Use the "known Taylor series" (Table 10.1) to determine the sum of the series 27 23 25 33.3 35.5 37.7 2/3 - + - +
5 Use the "known Taylor series" (Table 10.1) to determine the sum of the series 27 23 25 33.3 35.5 37.7 2/3 - + - +
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Hello,
Can someone clearly show all of the steps for solving problem 5?
![## Series and Limits
### For the next three exercises, use series to evaluate the limit.
1. \[
\lim_{{y \to 0}} \frac{{\arctan y - \sin y}}{{y^3 \cos y}}
\]
2. \[
\lim_{{h \to 0}} \frac{{(\sin h)/h - \cos h}}{{h^2}}
\]
3. \[
\lim_{{x \to \infty}} (x + 1) \sin \left( \frac{1}{{x + 1}} \right)
\]
### Additional Exercise
4. Determine a polynomial that will approximate
\[
F(x) = \int_{0}^{x} \arctan t \, dt
\]
throughout the interval \([0, 0.5]\) with an error of magnitude less than \(10^{-3}\).
### Taylor Series
5. Use the "known Taylor series" (Table 10.1) to determine the sum of the series
\[
\frac{2}{3} + \frac{2^3}{3^3 \cdot 3} + \frac{2^5}{3^5 \cdot 5} + \frac{2^7}{3^7 \cdot 7} + \ldots
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77ccc228-61ae-45bc-bff3-36e89ce5abe9%2Fca72c40b-7dfa-4075-9fa9-69b986edb194%2Fjtjbxnx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Series and Limits
### For the next three exercises, use series to evaluate the limit.
1. \[
\lim_{{y \to 0}} \frac{{\arctan y - \sin y}}{{y^3 \cos y}}
\]
2. \[
\lim_{{h \to 0}} \frac{{(\sin h)/h - \cos h}}{{h^2}}
\]
3. \[
\lim_{{x \to \infty}} (x + 1) \sin \left( \frac{1}{{x + 1}} \right)
\]
### Additional Exercise
4. Determine a polynomial that will approximate
\[
F(x) = \int_{0}^{x} \arctan t \, dt
\]
throughout the interval \([0, 0.5]\) with an error of magnitude less than \(10^{-3}\).
### Taylor Series
5. Use the "known Taylor series" (Table 10.1) to determine the sum of the series
\[
\frac{2}{3} + \frac{2^3}{3^3 \cdot 3} + \frac{2^5}{3^5 \cdot 5} + \frac{2^7}{3^7 \cdot 7} + \ldots
\]
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