Maclaurin series for a function Provided I have the

Kiara Haas

Kiara Haas

Answered question

2022-03-23

Maclaurin series for a function
Provided I have the function
f(x)=(1+x)1x,
and I want to calculate a 3rd order Maclaurin series, how can that be done without taking direct derivatives (as this seems hard..). I know that
(1+x)1x=eln(1+x)x,
and the Maclaurin series for ex is easy to prove, so I think it's a good direction..

Answer & Explanation

aznluck4u72x4

aznluck4u72x4

Beginner2022-03-24Added 16 answers

ln(1+x)x=1x2+x23x34+o(x3)
Substitute -x2+x23x34 to u in the development of eu, first computing the succesive powers of u:
u2=x24x33+o(x3)
u3x38+o(x3),
so that
(1+x)1x=e(1+u+u22+u36+o(u))
=e(1x2+1148x2716x3+o(x3)).

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