TITLE: Calculate the limit with the series of fractional powers: $\lim_{n\to\infty}\frac{1+{1.5}^{\frac1n}+{1.5}^{\frac2n}+\cdots+{1.5}^{\frac{n-1}n}}n$
QUESTION [8 upvotes]: Calculate the limit
$$\lim_{n\to\infty} \frac{1+{1.5}^{\frac{1}{n}}+{1.5}^{\frac{2}{n}}+\cdots+{1.5}^{\frac{n-1}{n}}}{n}$$
I got $1.23315\ldots$ by computer programming, but is there any way to get the exact result?

REPLY [8 votes]: The numerator is just a geometric series of common ratio $1.5^{1/n}$ and you are computing
$$\lim_{n\to\infty}\frac{1.5-1}{n(1.5^{1/n}-1)}=\frac{0.5}{\log 1.5}\approx1.233151731\cdots$$
by an alternative definition/property of the logarithm (or by L'Hospital).

REPLY [7 votes]: $$\lim _{n\to \infty} \frac {1+1.5^{\frac 1 n}+ 1.5^{\frac 2 n}+ \cdots + 1.5^{\frac {n-1} n}} n = \lim_{n\to\infty} \frac 1 n \sum_{k=1}^n 1.5^{(k-1)/n} \\
= \int_0^1 1.5^x \, dx = \frac {1.5^x}{\ln 1.5} \Bigg|_0^1 =\frac {0.5}{\ln 1.5} $$