Power series for derivatives a. Differentiate the Taylor series centered at 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. f(x)=ln(1+x)

Jaya Legge 2021-02-18 Answered
Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
f(x)=ln(1+x)
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Clara Reese
Answered 2021-02-19 Author has 120 answers

Taylor series of function f(x) at a is defined as:
f(x)=f(a)+f(a)1!(xa)+f(a)2!(xa)2+f(a)3!(xa)3+...
Consider the given:
f(x)=ln(1+x)
f(x)0+ddx(ln(1+x))(0)1!x+d2dx2(ln(1+x))(0)2!x2+d3dx3(ln(1+x))(0)3!x3+...
f(x)0+11!x+12!x2+23!x3+64!x4+245!x5+...
f(x)x12x2+13x314x4+15x5+...
b) The series sum representation-
n=1(1)n+1xnn
Apply ratio test and find interval of convergence.
If |an+1an|q eventually for some n=1|an| converges
If |an+1an|>1 eventually then n=1an diverges.
Consider the given series:
f(x)=n=1(1)n+1xnn
an=(1)n+1xnn
ak+1=(1)n+2xn+1n+1
an+1an=(1)n+2xn+1n+1(1)n+2xnn
=(1)nx(n+1)
L=limn|an+1an|
=limn|(1)x(1+1n)|
=|x|
From ratio test, if L > 1 series diverges.
L< 1 series converges.
||x||<1 series converges.
Find the interval of convergence. 1<x1
Interval of convergence ={x:x=1<x1}
Interval of convergence =(-1,1]

Not exactly what you’re looking for?
Ask My Question
Jeffrey Jordon
Answered 2021-12-16 Author has 2047 answers
Jeffrey Jordon
Answered 2021-12-16 Author has 2047 answers
Jeffrey Jordon
Answered 2021-12-17 Author has 2047 answers

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more