Abel's Limit Theorem/Examples

Examples of Use of Abel's Limit Theorem

Arbitrary Example 1

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.

Arbitrary Example 2

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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 $\sqrt {1 + x}$:

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

Arbitrary Example 3

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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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