Limit of a continuous product!
n → ∞ lim n n n ⎝ ⎛ k = 1 ∏ n k ! ⎠ ⎞ ⎝ ⎛ k = 1 ∏ n k k − 1 ⎠ ⎞ = ?
Give your answer to 1 decimal place.
The answer is 0.0.
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1 solution
Moderator note:
Your solution writeup could be improved.
You should start saying that you claim that k = 1 ∏ n k ! k = 1 ∏ n k k − 1 = ( n ! ) n and you're going to use induction to solve it.
Is there a way to solve it without induction?
I've expanded the double productory to the first fourth terms and noted the pattern. Then, just like you, I've demonstrated that the pattern follows, by induction. Finally, for a fixed n , the expansion of the factorial in the nummerator leaves the fraction n 1 as last term, which limit is zero, just like the entire product. Pretty good problem, indeed!
Yes there is a way By taking log both sides And then solving We can get easily the term
We claim that - k = 1 ∏ n k ! k = 1 ∏ n k k − 1 = ( n ! ) n and we'll use induction to verify it.