Hi~ Thank you for your help!

Question

I was trying to work on a problem about Taylor series, but i don't think im approaching the problem the right way.
I have to find the fifth order Taylor polynomial and Taylor series for the function f(x) at x = 0.
f(x) = 1/(x+2)
I tried to just write out the first and second and third derivatives, and this was fine, but then i got to the fourth and fifth derivatives and the quotient rule made the derivative look really confusing....is there a better way to approach this problem?

I might be doing this totally wrong since we just started learning Taylor series today in class, so any advice is much appreciated.

Answer 1

It turns out that if you had done your derivatives correctly, you would have found the rule:
for f(x)=1/(x+2)
f⁽ⁿ⁾(x) = (-1)⁽ⁿ⁾n!/(x+2)ⁿ⁺¹

I got f⁽⁵⁾(x)=-120/(x+2)⁶.

Hope that helps.