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(b) Show that I(n + 1) = n · I'(n), using integration by parts.

(b) Show that I(n + 1) = n · I'(n), using integration by parts.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 78E
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Use integration by parts for part b

4. The gamma function, which plays an important role in advanced applications, is defined for n >1 by I(n) = tn-le-t dt. (a) Show that the integral defining I'(n) converges for n > 1. (Hint: Show that t"-le-t <t-2 for t sufficiently large.) (b) Show that r(n + 1) = n · r(n), using integration by parts. (c) Show that I(n+1) = n! if n > 1 is an integer. Thus I(n) provides a way of extending the definition of n-factorial when n is not an integer.
Transcribed Image Text:4. The gamma function, which plays an important role in advanced applications, is defined for n >1 by I(n) = tn-le-t dt. (a) Show that the integral defining I'(n) converges for n > 1. (Hint: Show that t"-le-t <t-2 for t sufficiently large.) (b) Show that r(n + 1) = n · r(n), using integration by parts. (c) Show that I(n+1) = n! if n > 1 is an integer. Thus I(n) provides a way of extending the definition of n-factorial when n is not an integer.
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