Find the first 4 nonzero terms in the Series

Question

The function f(x)=ln⁡x has a Taylor series at a=8. Find the first 4 nonzero terms in the series, that is write down the Taylor polynomial with 4 nonzero terms.

Answer 1

Recall the Taylor Series formula

f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ... + fⁿ(a)(x-a)ⁿ/n! = ∑[n=0,∞] fⁿ(a)/n!•(x-a)ⁿ

Find f(a) and its first 3 derivatives at a=8

f(a) = ln(a) = ln(8)

f'(a) = 1/a = 1/8

f''(a) = -1/a² = -1/64

f'''(a) = 2/a³ = 2/512 = 1/256

Evaluate

f(x) ≈ ln(8) + (1/8)(x-8) - (1/64)(x-8)²/2! + (1/256)(x-8)³/3!

f(x) ≈ ln(8) + (x-8)/8 - (x-8)²/128 + (x-8)³/1536

Hope this helped!