Find the Taylor series generated by f at x = a.
-$$f ( x ) = \frac { 1 } { x ^ { 2 } } , a = 10$$
A) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } n ( x - 10 ) ^ { n } } { 10 ^ { n + 2 } }$
B) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } n ( x - 10 ) ^ { n } } { 10 ^ { n + 1 } }$
C) $\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } ( \mathrm { n } + 1 ) ( \mathrm { x } - 10 ) ^ { \mathrm { n } } } { 10 ^ { \mathrm { n } + 2 } }$
D) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( n + 1 ) ( x - 10 ) ^ { n } } { 10 ^ { n + 1 } }$

Question

Quizplus · Accepted Answer

The Taylor series of a function $f(x)$ at $x=a$ is given by $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$, where $f^{(n)}(a)$ is the $n$th derivative of $f$ evaluated at $x=a$. For $f(x) = \frac{1}{x^2}$, the $n$th derivative at $x=a$ involves a factor of $(-1)^n(n+1)$ and the denominator $a^{n+2}$ due to the pattern of differentiation of $x^{-2}$. Thus, the correct series is the one that matches this pattern, which is option C: $\sum_{n=0}^{\infty} \frac{(-1)^n(n+1)(x-10)^n}{10^{n+2}}$.

Find the Taylor Series Generated by F at X = f(x)=1x2,a=10f ( x ) = \frac { 1 } { x ^ { 2 } } , a = 10f(x)=x21​,a=10

Find the Taylor Series Generated by F at X = $f ( x ) = \frac { 1 } { x ^ { 2 } } , a = 10$