Taylor series

Question

Find the first five non-zero terms of Taylor series centered at x=6x=6 for the function below.
f(x)=lnx

 

What is the radius of convergence?

Answer 1

f(x) = ln x

 

df/dx = x^-1

 

by repeatedly applying the power rule:

 

dⁿf/dxⁿ = (-1)^n-1(n-1)! / xⁿ

 

f⁽ⁿ⁾(6) = (-1)^n-1(n-1)!/6ⁿ

 

a_n = (-1)^n-1 / (n6ⁿ)

Therefore:

 

ln x = ln 6 + (x-6)/6 - (x-6)²/72 + (x-6)³/648 - (x-6)⁴/5184 + ...

 

The radius of convergence:

 

|a_n+1/a_n| = (n-1)(x-6)/(6n)

 

This need this to approach a value < 1. This is the case if |x-6| < 6.

 

We at x = 0, the series diverges. At x = 12, we get the series converges by the alternating series test.

 

R = 6 ; Interval = (0,12]