I've been trying to puzzle out how to find this limit but I am well and truly stumped. I would really appreciate the help...
Find the limit as n goes to infinity of (e^(1/n)+e^(2/n)+e^(3/n)+........+e^(n/n))/n
Thanks!
I've been trying to puzzle out how to find this limit but I am well and truly stumped. I would really appreciate the help...
Find the limit as n goes to infinity of (e^(1/n)+e^(2/n)+e^(3/n)+........+e^(n/n))/n
Thanks!
HI ,
If you have learnt a topic called Riemann Sum , I advice to apply it to obtain the limit .
If you not or if you feel bored about this method , i suggest solving it in this way :
$\displaystyle \lim_{n\to\infty} \frac{1}{n} (e^{1/n} + e^{2/n} + e^{3/n} + ... + e^{n/n} ) $
$\displaystyle = \lim_{n\to\infty} \frac{1}{n} (t + t^2 + t^3 + ... + t^n ) $ where $\displaystyle t = e^{1/n} $
$\displaystyle = \lim_{n\to\infty} \frac{1}{n} \frac{t( t^n - 1)}{t-1} $
Since $\displaystyle \lim_{n\to\infty} e^{1/n} ( or ~ t) = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^{n/n} =\lim_{n\to\infty} (1 + \frac{1}{n} ) $
The limit is
$\displaystyle \lim_{n\to\infty} \frac{1}{n} (1+\frac{1}{n}) \frac{e-1}{1+\frac{1}{n} - 1 } $
$\displaystyle = \lim_{n\to\infty} \frac{1}{n} n(1+ \frac{1}{n})(e-1) $
$\displaystyle = \lim_{n\to\infty} (1+\frac{1}{n})(e-1) = e-1$