I need the interval of convergence. Find the Taylor Series of f(x)cos() centered at a= S. Find the interval of convergence. Show all necessary steps.= COS%3DMaximum file size: 250MB, maximu

Given fx=cosx3 about a=3π2. The Taylor series of a function, fx about x=a is given by,…

Answered: Find the Taylor Series of f(x) = c os()…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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I need the interval of convergence.

Find the Taylor Series of f(x)
cos() centered at a
= S. Find the interval of convergence. Show all necessary steps.
= COS
%3D
Maximum file size: 250MB, maximu

Expert Solution

Step 1

Given $f (x) = \cos (\frac{x}{3})$ about $a = \frac{3 π}{2}$ .

The Taylor series of a function, $f (x)$ about $x = a$ is given by,

$f (x) = f (a) + (x - a) f' (a) + \frac{{(x - a)}^{2}}{2!} f " (a) + \dots \dots$

Differentiating $f (x) = \cos (\frac{x}{3})$ , with respect to x,

$\begin{array}{rcl} f (x) & = & \cos (\frac{x}{3}) \\ f' (x) & = & - \frac{1}{3} \sin (\frac{x}{3}) \\ f " (x) & = & - \frac{1}{3^{2}} \cos (\frac{x}{3}) \\ f^{(3)} (x) & = & \frac{1}{3^{3}} \sin (\frac{x}{3}) \\ ⋮ \end{array}$

Substitute, $x = \frac{3 π}{2}$ in the above equation,

$\begin{array}{rcl} f (\frac{3 π}{2}) & = & 0 \\ f' (\frac{3 π}{2}) & = & - \frac{1}{3} \\ f " (\frac{3 π}{2}) & = & 0 \\ f^{(3)} (\frac{3 π}{2}) & = & \frac{1}{3^{3}} \\ ⋮ \end{array}$

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