Abel's Limit Theorem/Examples/Arbitrary Example 3

Examples of Use of Abel's Limit Theorem

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Let:

$\map g x = \dfrac 1 {1 + x^2}$

which is differentiable for all real $x$.

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When $\size x < 1$, $\map g x = \ds \sum_{n \mathop \ge 0} \paren {-1}^n x^{2 n}$ by expanding a geometric series.

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While $\map g x$ has a limit as $x \to 1^{-}$ (namely $1/2$), the power series does not converge at $x = 1$.

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