Taylor series of f(x)=ln(x+1) centred at 2

[jmher0403]

Homework Statement

Taylor series of f(x)=ln(x+1) centred at 2

Homework Equations

from 0 to infinity ∑ cn(x-a)n

cn = f(n)(a)/n!

The Attempt at a Solution

f(x) = ln(1+x)
f'(x) = 1/(1+x)
f''(x) = -1/(1+x)2
f'''(x) = 2/(1+x)3
f''''(x) = -6/(1+x)4f(2) = ln(3) = 1.0986
f'(2) = 1/3
f''(2) = -1/9
f'''(2) = 2/27
f''''(2) = -6/81

f(2)+f'(2)(x-2)+f''(2)(x-2)²/2!+f'''(2)(x-2)³/3!+f''''(2)(x-2)⁴/4!

I can't see any pattern except partially ..(-1)^n-1(x-2)ⁿ/3ⁿn!

I have no idea what to do with ln3

Please help!

 

[cjc0117]

Just write the ln3 outside of the sum. Then try to think of a pattern for the remaining terms. First, start by trying to find a general formula for f^{(n)}(2).