- #1
naaa00
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Homework Statement
Use the binomial theorem to rpove that for n a positive integer we have:
(1 + 1/n)^n = 1 + sum(k=1 to n) [1/k! product(r=0 to k-1) (1 - r/n)]
The Attempt at a Solution
(1 + 1/n)^n = 1 + sum(k=1 to n) (n choose r) 1^n-k (1/n)^k, where (n choose r) = n!/r!(n - r)!, the binomial coefficients.
I'm trying to fit "n!/r!(n - r)!" to an expression that involves the products, since product(k=1 to n) n = n!
The product on the RHS I rewrite it as: product(r=0 to k-1) [(n - r)/n]
[product(r=0 to k-1) (n - r)] x [product(r=0 to k-1) (1/n)]
=> (n - r)! (1/n)!
So...
(1 + 1/n)^n = 1 + sum(k=1 to n) [1/k! (n-r)! (1/n)!] or (1 + 1/n)^n = 1 + sum(k=1 to n) [(n-r)!/k!n!]
I don't like this. I feel that all of this is taking me nowhere.
Any ideas will be very appreciated.