Is there any expansion for log(1+x) when x>1?

Trace Glass 2022-10-24 Answered
Is there any expansion for log ( 1 + x ) when x > 1?
Is there any expansion for log ( 1 + x ) when x > 1?
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Answers (2)

bargeolonakc
Answered 2022-10-25 Author has 16 answers
You can expand the function log ( 1 + x ) around any point at which it is defined. This means there exists an expansion of log ( 1 + x ) around the point x = 2, for example, however it will be of the form
log ( 1 + x ) = i = 1 a i ( x 2 ) i ,
and the expansion will be valid for x ( 1 , 5 )
However, I imagine you want the expansion to of the form
log ( 1 + x ) = i = 1 a i x i ,
for which you will have a problem. The problem is that any power series
i = 1 a i ( x x 0 ) i
will converge on a interval symmetric around x 0 (meaning an interval of the type ( x 0 δ , x 0 + δ )). This means that if the series expansion for log ( 1 + x ) will converge for x > 1, it will also converge for some x < 1 which is impossible.
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Deja Bradshaw
Answered 2022-10-26 Author has 4 answers
You are looking for Laurent series for | x | > 1
ln ( 1 + x ) = ln ( x ( 1 + 1 / x ) ) = ln ( x ) + ln ( 1 + 1 / x ) = ln ( x ) + k = 1 ( 1 ) k k x k
with the condition of convergence | x | > 1
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