Question

Taylor seriesb. Write the power series using summation notation.f(x)=sin x, a = π/2

Accepted Answer

Recall the Taylor series formula for a function $$ f(x) $$ centered at $$ a $$:

$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n $$

where $$ f^{(n)}(a)  is $$the $$ n -th $$derivative of $$ f $$ evaluated at $$ a . $$Identify the function and center: here, $$ f(x) = \sin x $$ and $$ a = \frac{\pi}{2} . We $$need to find the derivatives of $$ \sin x $$ evaluated at $$ x = \frac{\pi}{2} . $$Calculate the first few derivatives of $$ \sin x $$ and evaluate them at $$ x = \frac{\pi}{2} $$:

- $$ f(x) = \sin x $$
- $$ f'(x) = \cos x $$
- $$ f''(x) = -\sin x $$
- $$ f^{(3)}(x) = -\cos x $$
- $$ f^{(4)}(x) = \sin x $$

Evaluate each at $$ x = \frac{\pi}{2} $$:

- $$ f\left(\frac{\pi}{2}\right) = 1 $$
- $$ f'\left(\frac{\pi}{2}\right) = 0 $$
- $$ f''\left(\frac{\pi}{2}\right) = -1 $$
- $$ f^{(3)}\left(\frac{\pi}{2}\right) = 0 $$
- $$ f^{(4)}\left(\frac{\pi}{2}\right) = 1 $$ Notice the pattern in the derivatives evaluated at $$ a = \frac{\pi}{2} $$: the values cycle through $$ 1, 0, -1, 0, 1, \ldots . $$This pattern will help simplify the summation. Write the Taylor series in summation notation using the pattern found:

$$ \sin x = \sum_{n=0}^{\infty} \frac{f^{(n)}\left(\frac{\pi}{2}\right)}{n!} (x - \frac{\pi}{2})^n $$

Substitute the values of the derivatives into the summation to express the power series explicitly.

Taylor series

b. Write the power series using summation notation.

f(x)=sin x, a = π/2

Verified video answer for a similar problem:

Key Concepts

Taylor Series Expansion

Derivatives of sin(x)

Summation Notation for Power Series