Question

Using \(f(x) = \sin x\), construct the power series

$\sum _{K=0}^{\infty }\frac{f^k\left(0\right)}{k!}{x}^k$

Short answer

The power series of the given function is :
$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+.........$

Step-by-step solution

Given Information

Given function is \(f(x) =\sin x\).

Given power, series is to be constructed

\(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...\)

Here given form of power series is a Maclaurin series as the point at which the series is to be defined is zero.

\(\oint_{a}^{b}x^{3}-x^{2}+\frac{4}{9}x\)

Finding Maclaurin series of \(f(x)=\)sinx

Maclaurin series form is given by:

$\frac{f\left(0\right)}{0!}{x}^0+\frac{f^1\left(0\right)}{1!}{x}^1+\frac{f^2\left(0\right)}{2!}{x}^2+........$

Here, f(x) = sin(x)

$f\left(0\right)=\sin \left(0\right)=0$

$f^1\left(0\right)=\cos \left(0\right)=1$

$f^2\left(0\right)=-\sin \left(0\right)=0$

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Substituting the above values in Maclaurin's general equation, we get:

$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+.......$