[math-fun] Limit question
What is lim ((1/n)^n + (2/n)^n + ... + (n/n)^n) n—>oo (exactly, if possible) ??? My bag of limit tricks doesn't readily seem to include how to solve this one. —Dan
The sum is dominated by the final terms, the initial terms vanishing in the limit. The final term is 1. The penultimate term is (1 - 1/n)^n, which becomes exp(-1). The preceding term is (1 - 2/n)^n, which becomes exp(-2). So the sum becomes 1 + exp(-1) + exp(-2) + ... = 1/(1 - exp(-1)). -- Gene On Sunday, December 17, 2017, 1:04:58 PM PST, Dan Asimov <dasimov@earthlink.net> wrote: What is lim ((1/n)^n + (2/n)^n + ... + (n/n)^n) n—>oo (exactly, if possible) ??? My bag of limit tricks doesn't readily seem to include how to solve this one. —Dan
Well, its continued fraction seems to be 1,1,1,2,1,1,4,1,1,6,1,1,8,1,1,... On 17-Dec-17 16:04, Dan Asimov wrote:
What is
lim ((1/n)^n + (2/n)^n + ... + (n/n)^n) n—>oo
(exactly, if possible) ???
My bag of limit tricks doesn't readily seem to include how to solve this one.
—Dan
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Dan Asimov -
Eugene Salamin -
Mike Speciner