What is Logarithmic Series Expansion ?

What is Logarithmic Series Expansion ?

Logarithmic Series

Definition

An expansion for loge (1 + x) as a series of powers of x which is valid only, when |x|<1.

Expansion of logarithmic series

Expansion of loge (1 + x) if |x|<1 then
What is Logarithmic Series Expansion 1
Replacing x by −x in the logarithmic series, we get
What is Logarithmic Series Expansion 2

Some Important results from logarithmic series

What is Logarithmic Series Expansion 3
(2) The series expansion of loge (1 + x) may fail to be valid, if |x| is not less than 1. It can be proved that the logarithmic series is valid for x = 1. Putting x = 1 in the logarithmic series.
We get,
What is Logarithmic Series Expansion 4
(3) When x = −1, the logarithmic series does not have a sum. This is in conformity with the fact that log(1 – 1) is not a finite quantity.

Difference between the exponential and logarithmic series

What is Logarithmic Series Expansion 5
(2) In the exponential series the denominator of the terms involve factorial of natural numbers. But in the logarithmic series the terms do not contain factorials.
(3) The exponential series is valid for all the values of x. The logarithmic series is valid, when |x|< 1.

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