Question regarding logarithms 2 What is ln(−1)? And would there a taylor series for ln(1+x^m)/(1-x^m)?

independanteng

independanteng

Answered question

2022-10-28

Question regarding logarithms 2
What is ln ( 1 )? And would there a taylor series for
ln 1 + x m 1 x m ?

Answer & Explanation

honotMornne

honotMornne

Beginner2022-10-29Added 12 answers

The most beautiful formula in mathematics is
e i π + 1 = 0
so the result as given by barak manos while I was typing.
With regard to the second question, consider
log ( 1 + y ) = i = 1 ( 1 ) i 1 y i i
log ( 1 y ) = i = 1 y i i
and now subtract to get
log 1 + y 1 y = 2 i = 1 y 2 i 1 2 i 1
Now, you can replace y by x m
Jack Ingram

Jack Ingram

Beginner2022-10-30Added 2 answers

Use Euler's Identity:
e i π + 1 = 0 1 = e i π ln ( 1 ) = ln ( e i π ) = i π

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