Beta Question Problem

Asked May 29 '18 at 23:09

Active May 29 '18 at 23:55

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I was trying to understand a problem relating to the beta function. In the middle of the solution, there is this line that confuses me, as I have no idea how the infinite summation simplifies to the natural logarithm.

$$\int_0^1\sum_{n=1}^{\infty}\frac{\left(x\left(1-x\right)\right)^{\left(n-1\right)}}{2n}dx=\frac{1}{2}\int_0^1\ln\left(1-x\left(1-x\right)\right)\left(x\left(1-x\right)\right)^{-1}dx$$

edited May 29 '18 at 23:55

Robbie

asked May 29 '18 at 23:09

John A.

1

Do you know the Taylor series for $\ln(1-u)$? – Kenny Wong May 29 '18 at 23:11
@KennyWong Yes, but that has alternating signs (negative and positive), so I don't see how the author simplifies the integral using the Taylor series. – John A. May 30 '18 at 00:17
Sorry, I just fumbled, I think I get it. – John A. May 30 '18 at 00:20
@KennyWong Thanks for the help, I got it!!! – John A. May 30 '18 at 01:06

Beta Question Problem

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