Logarithmic Series

If x is a real number such that |x|<1, then the series is called the logarithmic series. It can be proved mathematically that this logarithmic series has the sum equal to log(1 + x). If x is a real number such that |x|<1, then the series is called the logarithmic series. It can be proved mathematically that this logarithmic series has the sum equal to log(1 + x).

Theorem:

Proof:

Since |x|<1, we have by binomial theorem,

Comparing, the coefficients of y in (1) and (2), we get

Remark:

The terms in log(1 + x) carry alternatively positive and negative signs. The terms in log(1 + x) do not involve factorials as in the exponential series.