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41. For which function is ∑n=0∞(−1)nx2n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}...
Jan 27, 2024
41. For which function is ∑n=0∞(−1)nx2n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !} the Taylor series about 0 ? (A) exe^{x} (B) e−xe^{-x} (C) sin⁡x\sin x (D) cos⁡x\cos x (E) ln⁡(1+x)\ln (1+x)
Generated Graph
Solution by Steps
step 2
This series is recognized as the Taylor series expansion of the cosine function, cos⁡(x)\cos(x), about 0
step 3
The general form of the Taylor series for cos⁡(x)\cos(x) is ∑n=0∞(−1)nx2n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!}
step 4
Therefore, the function for which the given series is the Taylor series about 0 is cos⁡(x)\cos(x)
D
Key Concept
Taylor Series of Cosine Function
Explanation
The Taylor series of cos⁡(x)\cos(x) about 0 is ∑n=0∞(−1)nx2n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!}, which matches the given series.
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