Use the definition to find the Taylor series centered at $c = \frac { \pi } { 4 } \text { for the function }$ $f ( x ) = \cos x$
A)
$$\cos x = \frac { \sqrt { 2 } } { 2 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \frac { n ( n - 1 ) } { 2 } } \left( x + \frac { \pi } { 4 } \right) ^ { n } } { n ! }$$
B)
$$\cos x = \frac { \sqrt { 2 } } { 2 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \frac { n ( n + 1 ) } { 2 } } \left( x - \frac { \pi } { 4 } \right) ^ { n } } { n ! }$$
C)
$$\cos x = \sqrt { 2 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } \left( x - \frac { \pi } { 4 } \right) ^ { n } } { n ! }$$
D)
$$\cos x = \sqrt { 2 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \frac { n ( n - 1 ) } { 2 } } \left( x + \frac { \pi } { 4 } \right) ^ { n } } { n ! }$$
E)
$$\cos x = \frac { \sqrt { 2 } } { 2 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } \left( x - \frac { \pi } { 4 } \right) ^ { n } } { n ! }$$

Question

Use the definition to find the Taylor series centered at $c = \frac { \pi } { 4 } \text { for the function }$ $f ( x ) = \cos x$
A)
$$\cos x = \frac { \sqrt { 2 } } { 2 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \frac { n ( n - 1 ) } { 2 } } \left( x + \frac { \pi } { 4 } \right) ^ { n } } { n ! }$$
B)
$$\cos x = \frac { \sqrt { 2 } } { 2 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \frac { n ( n + 1 ) } { 2 } } \left( x - \frac { \pi } { 4 } \right) ^ { n } } { n ! }$$ 
C)
$$\cos x = \sqrt { 2 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } \left( x - \frac { \pi } { 4 } \right) ^ { n } } { n ! }$$
D)
$$\cos x = \sqrt { 2 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \frac { n ( n - 1 ) } { 2 } } \left( x + \frac { \pi } { 4 } \right) ^ { n } } { n ! }$$
E)
$$\cos x = \frac { \sqrt { 2 } } { 2 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } \left( x - \frac { \pi } { 4 } \right) ^ { n } } { n ! }$$

Quizplus · Accepted Answer

The Taylor series of $\cos x$ centered at $c = \frac{\pi}{4}$ involves terms of $(x - \frac{\pi}{4})^n$, reflecting shifts from the center point $c$. The factor $\frac{\sqrt{2}}{2}$ is the value of $\cos(\frac{\pi}{4})$, which is the first term of the series when $n=0$. The pattern of signs, alternating due to the cosine function's derivatives, is correctly represented by the exponent $\frac{n(n+1)}{2}$ in option B, which ensures the correct alternation of signs for each term in the series.

Use the Definition to Find the Taylor Series Centered At c