Math Problem Statement
Evaluate the Maclaurin series.
[\displaystyle\pi -\frac{{{\pi }^{3}}}{3!}+\frac{{{\pi }^{5}}}{5!}-\unicode{0x2026}+{{\left( -1 \right)}^{n}}\frac{{{\pi }^{2n+1}}}{\left( 2n+1 \right)!}+\unicode{0x2026}]
Choose 1 answer:
Choose 1 answer:
(Choice A)
[0]
A
[0]
(Choice B)
[1]
B
[1]
(Choice C)
[\cos ( \pi^2 )]
C
[\cos ( \pi^2 )]
(Choice D)
[-\cos ( \pi^2 )]
D
[-\cos ( \pi^2 )]
Solution
The given series is:
This series is a Maclaurin series (Taylor series at ) for a known function. Specifically, it matches the Maclaurin series for the sine function:
By comparing the series, we can see that the expression given is the sine function evaluated at :
We know from trigonometry that:
Therefore, the correct answer is:
(Choice A) 0
Would you like more details or have any questions? Here are 5 related questions that could further your understanding:
- What is the general formula for the Maclaurin series of a function?
- How do you determine the radius of convergence for a Maclaurin series?
- Can you derive the Maclaurin series for the cosine function?
- What are the key differences between the Maclaurin and Taylor series?
- How does the sine function's periodicity affect its Maclaurin series?
Tip: When evaluating a series, try to recognize if it matches a known function's series expansion. This can simplify the problem significantly.
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Math Problem Analysis
Mathematical Concepts
Maclaurin series
Taylor series
Trigonometry
Formulas
Maclaurin series formula
Taylor series formula
Sine function series expansion
Theorems
Trigonometric identities
Suitable Grade Level
Advanced High School