Crazy Sum.
n = 1 ∑ ∞ 2 n n ! ∏ r = 0 n − 1 ( π − r ) = ( B A ) π − C
If the equation above holds true for positive coprime integers A , B , and C , enter the value of A + B − C .
The answer is 4.
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2 solutions
Can we use combination over irrational numbers like π as n should be an integer?
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Indeed, we can. This is known as the generalized binomial coefficient, which has the definition ( k α ) = k ! α ( α − 1 ) ( α − 2 ) . . . ( α − k + 1 ) for any arbitrary number (negative, real or even complex) α and positive integer k .
Is the answer (3/2)^pi - 1?
I got this by using Gamma functions
Write full solutions please.
I also got by gamma function
This problem can be solved easily using binomial series, as follows. ( 1 + x ) π = = n = 0 ∑ ∞ ( n π ) x n 1 + n = 1 ∑ ∞ n ! ∏ r = 0 n − 1 ( π − r ) x n . The answer follows by substituting x = 2 1 .