Abel's Limit Theorem/Examples/Arbitrary Example 1

Examples of Use of Abel's Limit Theorem

Let $\ds \map g x = \sum_{n \mathop \ge 1} \paren {-1}^{n - 1} \dfrac {x^n} n$ for $\size x < 1$.

Then:

$\map g x = \map \ln {1 + x}$

for $\size x < 1$.

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The series $\map g 1$ converges by Alternating Series Test,

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so by Abel's Limit Theorem:

$\map g 1 = \ds \lim _{x \mathop \to 1^{-} } \map g x = \lim_{x \mathop \to 1^{-} } \map \ln {1 + x} = \ln 2$

since the logarithm is a continuous function.