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Question Number 192791 by Mastermind last updated on 27/May/23

If n is a positive integer, prove that

2^{n} Γ (n + \frac{1}{2}) = 1.3.5 \dots (2 n - 1) \sqrt{π} .

Help!

Answered by Mathspace last updated on 27/May/23

m o n t r o n s q u e Γ (n + \frac{1}{2}) = \frac{1.3.5 \dots (2 n - 1)}{2^{n}} \sqrt{π}

p a r r e c u r e n c e s u r n

n = 1 Γ (\frac{3}{2}) = \frac{\sqrt{π}}{2}

v r a i e c a r

Γ (\frac{3}{2}) = Γ (1 + \frac{1}{2}) = \frac{1}{2} Γ (\frac{1}{2}) = \frac{\sqrt{π}}{2}

s u p p o s o n s l a r e l a t i o n v r a i e p o u r n

o n a Γ (n + 1 + \frac{1}{2})

= Γ (1 + n + \frac{1}{2}) = (n + \frac{1}{2}) Γ (n + \frac{1}{2})

= (n + \frac{1}{2}) \times \frac{1.3.5 \dots (2 n - 1)}{2^{n}} \sqrt{π}

= \frac{1.3.5 \dots . (2 n - 1) (2 n + 1)}{2^{n + 1}} \sqrt{π}

l a r e l a t i o n e s t v r a i e a l o r d r e

n + 1 .

Commented by Mastermind last updated on 28/May/23

you mean it should be (2 n + 1) not (2 n - 1) ?

besides pls use english