Difference between revisions of "009C Sample Final 2, Problem 6"
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| − | | | + | |The power series of <math style="vertical-align: -1px"> \sin x</math> is <math>\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}.</math> |
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| − | | | + | |So, the power series of <math style="vertical-align: -5px">\sin(x^2)</math> is |
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| − | | | + | | <math>\begin{array}{rcl} |
| + | \displaystyle{} & = & \displaystyle{}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{} | ||
| + | \end{array}</math> | ||
|} | |} | ||
Revision as of 09:59, 5 March 2017
(a) Express the indefinite integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \sin(x^2)~dx} as a power series.
(b) Express the definite integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^1 \sin(x^2)~dx} as a number series.
| Foundations: |
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| What is the power series of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x?} |
| The power series of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}.} |
Solution:
(a)
| Step 1: |
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| The power series of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}.} |
| So, the power series of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(x^2)} is |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{} & = & \displaystyle{}\\ &&\\ & = & \displaystyle{}\\ &&\\ & = & \displaystyle{} \end{array}} |
| Step 2: |
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(b)
| Step 1: |
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| Step 2: |
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| Final Answer: |
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| (a) |
| (b) |