Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Q-30E (page 435)

Question: In Problems 29–34, determine the Taylor series about the point X0for the given functions and values of X0.

30.f(x)=x-4x0=1

Short Answer

Expert verified

The required expression is .∑n-0∞(-1)n(x-1)

Step by step solution

01

Taylor series

For a function the Taylor series expansion about a point is given by,f(x-x0)=f(x0)+f'(x0).(x-x0)+f''(x0).(x-x0)22!+f'''(x0)(x-x0)33!+....

02

Derivatives of function at x0.

We have to calculate the Taylor series expansion for, f (x) = x-1 at x0=1.

Calculating the derivatives of function at x0.

f(x)=x-1thenf(x0)=1

f'(x)=x-2thenf'(x0)=-1

f''(x)=2x-3thenf''(x0)=2

f'''(x)=6x-4thenf'''(x0)=-6

f''''(x)=24x-5thenf''''(x0)=24

03

Substitute the derivatives in Taylor series

Substituting the above derivatives in Taylor series expansion for the function at x0=1, then,

(x-1)-1=1-1(x-1)+2.(x-1)22!-6(x-1)33!+24.(x-1)44!+....=1-(x-1)+(x-1)2-(x-1)3+(x-1)4+....=∑n-0∞(-1)n.(x-1)n

Hence, the required expression is=∑n-0∞(-1)n.(x-1)n

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Question: In Problems 1–10, determine all the singular points of the given differential equation.

9. (sin)y"-(in)y=0

Question: In Problems 1–10, determine all the singular points of the given differential equation.

3.θ2-2y"+2y'+(sinθ)y=0

In Problems 1-10, use a substitution y=xr to find the general solution to the given equation for x>0.

d2y/dx2=5/x dy/dx-13/x2 y

Question: Let

Show that fn(0)=0for n=0,1,2....and hence that the Maclaurin series for f(x)is 0+0+0+...., which converges for all xbut is equal to f(x)only when x=0. This is an example of a function possessing derivatives of all orders (at x0=0), whose Taylor series converges, but the Taylor series (about x0 =0) does not converge to the original function! Consequently, this function is not analytic at x=0.

Question: In Problems 17-20, find a power series expansion for f'(C), given the expansion for f(x).

17.fx=1+x-1=∑n=0m(-1nxn
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free