The series 1-x^(2)+(x^(4))/(2!)-(x^(6))/(3!)+(x^(8))/(4!)+cdots+(-1)^(n)(x^(2n))/(n!)+cdots converges to which of the following? (A) cos(x^(2))+sin(x^(2)) (B) 1+x sin x (C) cos x (D) e^(-x^(2))

Identify general term: Identify the general term of the series. The general term of the series is (-1)^(n)(x^(2n))/(n!). This resembles the Taylor series expansion of e^(x) but with x replaced by -x^2. Recognize represented function: Recognize the function represented by the series. The series is the Taylor series for e^(-x^2), since replacing x in the e^(x) series with -x^2 gives us the terms of the given series. Match series to function: Match the series to the correct function from the options. The series matches the Taylor series for e^(-x^2), which is option (D).

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Calculus

Find derivatives of using multiple formulae

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Q. The series

1-x^{2}+\frac{x^{4}}{2!}-\frac{x^{6}}{3!}+\frac{x^{8}}{4!}+\cdots+(-1)^{n} \frac{x^{2 n}}{n!}+\cdots

converges to which of the following?

\newline

(A)

\cos \left(x^{2}\right)+\sin \left(x^{2}\right)

\newline

(B)

1+x \sin x

\newline

(C)

\cos x

\newline

(D)

e^{-x^{2}}

Identify general term: Identify the general term of the series. $\newline$ The general term of the series is $(-1)^n(x^{2n})/(n!)$ . This resembles the Taylor series expansion of $e^{x}$ but with $x$ replaced by $-x^2$ .
Recognize represented function: Recognize the function represented by the series. The series is the Taylor series for $e^{-x^2}$ , since replacing $x$ in the $e^{x}$ series with $-x^2$ gives us the terms of the given series.
Match series to function: Match the series to the correct function from the options. $\newline$ The series matches the Taylor series for $e^{(-x^2)}$ , which is option $(D)$ .