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application of logarithm series
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(Application)
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The integrand of the improper integral
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(1) |
is not defined at the lower limit 0. However, from the Taylor series expansion $$\ln(1\!+\!x) \;=\; x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+-\ldots \qquad (-1 < x \leqq 1)$$ of the natural logarithm we obtain the expansion of the integrand $$ \frac{\ln(1\!+\!x)}{x} \;=\; 1-\frac{x}{2}+\frac{x^2}{3}-\frac{x^3}{4}+-\ldots \qquad (-1 < x < 0,\;\; 0 < x \leqq 1) $$ whence
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(2) |
This implies that the integrand of (1) is bounded on the interval $[0,\,1]$ and also continuous, if we think that (2) defines its value at $x = 0$ . Accordingly, the integrand is Riemann integrable on the interval, and we can determine the improper integral by integrating termwise:
By the entry on Dirichlet eta function at 2, the sum of the obtained series is $\eta(2) = \frac{\pi^2}{12}$ . Thus we have the result
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(3) |
Similarly, using the series $$\ln(1\!-\!x) \;=\; -x-\frac{x^2}{2}-\frac{x^3}{3}-\frac{x^4}{4}-\ldots \qquad (-1 \leqq x < 1)$$ and the result in the entry Riemann zeta function at 2, one can calculate that
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(4) |
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"application of logarithm series" is owned by pahio.
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Cross-references: calculate, series, sum, Riemann integrable, continuous, interval, bounded, implies, natural logarithm, Taylor series, lower limit, improper integral, integrand
There are 2 references to this entry.
This is version 8 of application of logarithm series, born on 2009-05-17, modified 2009-05-18.
Object id is 11790, canonical name is ApplicationOfLogarithmSeries.
Accessed 615 times total.
Classification:
| AMS MSC: | 33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions) |
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Pending Errata and Addenda
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