Find Taylor series of function

Question

Find Taylor series of function f(x)= ln(x) at a=9

f(x) = Sum (n=0 to inf)

C0=

C1=

C2=

C3=

C4=

Find the interval of convergence.

The series is convergent:

from x= __ left end, included (yes or no)

to x=__ right end, included (yes or no)

Answer 1

Given f(x) = ln x Therfefore, f⁽¹⁾ = 1/x , f⁽²⁾ = -1/x², f⁽³⁾ = 2/x³, and f⁽⁴⁾ = -6/x⁴ Here f⁽ⁱ⁾ is i^th derivative f(x)

Taylor's series centered at 9, a = 9.

f(x) = ln x = ln 9 + (x-9)×1/9 + (-1)¹ × (x-9)²/(2!×9^2) + (-1)² ×2 (x-9)³/(3!×9^3) + (-1)³ × 6 (x-9)⁴/(4!×9^4)

= 2.197 + (x-9)/9 = (x-9)²/(2×9^2) + (x-9)³/(3×9^3) - (x-9)⁴/(4×9^4)

C_n = f⁽ⁿ⁾(a)/n! gives C₀ = 2.197, C₁ = 1/ (1!×9^1), C₂ = 2/(2!×9^2), C₃ = 2/(3!×9^3), and C₄ = 6/(4!×9^4)

Radius of convergence R = 9

from x= _9_ left end, included (yes or no) Yes

to x=_9_ right end, included (yes or no) Yes