Abel's Limit Theorem/Examples/Arbitrary Example 2

Examples of Use of Abel's Limit Theorem

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Let:

$\ds \map g x = \sum_{n \mathop \ge 0} \frac {\paren {-1}^{n - 1} \paren {2 } !} {2^{2 n} n!^2 \paren {2 n - 1} } x^n$

for $\size x < 1$.

Then:

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

for $\size x < 1$.

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The series $\map g 1$ is absolutely convergent

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so by Abel's Limit Theorem and the continuity of $\sqrt {1 + x}$:

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