Taylor Series question
Let f(x) = ln(1+x)
Find a formula for the nth Taylor Polynomial of f, and then calculate p_n(1).

Ok so I got up to the second derivative. Then for the third derivate, my response was
2/((1+x)^3)

Why is the third derivative 2!/((1+x)^3).. where does the factorial come from?

Question

Taylor Series question
Let f(x) = ln(1+x) 
Find a formula for the nth Taylor Polynomial of f, and then calculate p_n(1).

Ok so I got up to the second derivative. Then for the third derivate, my response was
2/((1+x)^3)

Why is the third derivative 2!/((1+x)^3).. where does the factorial come from?

tkhunny · Answer

Hard way!

$f(x) = \ln(1+x)$

$f'(x) = \dfrac{1}{1+x} = 1 - x + x^{2} - x^{3} + ...$

$f(x) = x - \dfrac{x^{2}}{2} + \dfrac{x^{3}}{3} - \dfrac{x^{4}}{4} + ...$

The funny factorial in the numerator is just to offset almost all of the factorial you would normally find in the denominator.

anonymous · Answer

i see.. so when we take the the third derivative.. instead of writing 6/((1+x)^4) we write 3!.. so when solving for p_n(x) it cancels out.. got it.. thank you!

tkhunny · Answer

Good work.  You just have to build it.

anonymous · Answer

Thank you tkhunny!!