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Single Variable Calculus

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Taylor Series of the Square Root Function

SVCALC-V41U1L

Let $\sum_{n=0}^\infty c_n (x-4)^n$ be the Taylor series of $f(x)=\sqrt{x}$ at $x=4$.

Which of the following expression gives the $n$th coefficient $c_n$ for $n\geq2$?

A

$\cfrac{(-1)^{n+1}1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2^{2n}n!}$

B

$\cfrac{(-1)^{n+1}1\cdot3\cdot5\cdot\ldots\cdot(2n-3)}{2^{2n-1}n!}$

C

$\cfrac{(-1)^{n+1}1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2^{2n-1}n!}$

D

$\cfrac{(-1)^{n+1}1\cdot3\cdot5\cdot\ldots\cdot(2n-3)}{2^{3n-1}n!}$

E

$\cfrac{(-1)^{n+1}1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2^{3n-1}n!}$