Find the taylor series of ln(1+x)

[vande060]

Homework Statement

find the taylor series of ln(1+x) centered at zero

Homework Equations

from 0 to infinity ∑ c_n(x-a)ⁿ

c_n = 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)⁴


f(0) = 0
f'(0) = 1
f''(0) = -1
f'''(0) = 2
f''''(0) = -6

^ I feel like there is a mistake somewhere in the drivative, because one i set out the taylor series, it doesn't make much sense to me

0 + x/1! -x²/2! +2x³/3! -6x⁴/4!

 

[ideasrule]

There's no mistake. Can you see a pattern? Can you find the general equation for the nth term in the expansion?

 

[vande060]

There's no mistake. Can you see a pattern? Can you find the general equation for the nth term in the expansion?

I tried to develop an nth term, but the 2 and 6 are messing me up, that why I thought it was wrong.

general term xⁿ(-1)ⁿ⁺¹/n! starting at n=1

It should resemble something like that, but I am not too sure, again I cannot figure out how to develop a general term that will explain the 2 and 6

 

[ideasrule]

Note that 2=2*1 and 6=3*2*1. I bet the next coefficient will be 24=4*3*2*1 (check this!)

 

[SammyS]

2=2!

6=3!

Look at the next couple of derivatives of f if you need to.

 

[vande060]

okay I got it thanks