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How can I find the range of the parameter a such that the improper integral

Integrate[(x^(a - 1))/(1 + x), {x, 0, ∞}]

converges?

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2 Answers 2

$Version
"9.0 for Mac OS X x86 (64-bit) (January 24, 2013)"
Integrate[(x^(a - 1))/(1 + x), {x, 0, Infinity}]
ConditionalExpression[Pi*Csc[a*Pi], 0 < Re[a] < 1]
Assuming[0 < Re[a] < 1, Integrate[(x^(a - 1))/(1 + x), {x, 0, Infinity}]]
Pi*Csc[a*Pi]
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Since there are no path singularities and the behavior is monotonic, you you in effect reverse the integral test on infinite series. That is, find values of a for which the corresponding infinite sum will converge. This is done with SumConvergence.

SumConvergence[(x^(a - 1))/(1 + x), x]

(* Out[9]= Re[a] < 1 *)

--- edit ---

That said, I prefer the answer by @Bob Hanlon. In particular it captures both bounds, one of which is necessitated by behavior at the origin rather than at infinity. To do that with my method one might use the substitution x->1/y, dx->-1/y^2*dy so as to in effect move the origin to infinity.

SumConvergence[((1/x)^(a - 1))/(1 + 1/x)/x^2, x]

(* Out[59]= Re[a] > 0 *)

--- end edit ---

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