I am sort of confused.
Suppose we are given the series,
$\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n} \frac{k^{99}}{n^{100}}$
How can this be written as an integral, and what would the variable be?
In this series given, which terms are the constants? Is it $n^{100}$??
Wouldn't the above be written as,
$\displaystyle \lim_{n\to\infty} \frac{1}{n^{99}} \cdot \frac{1}{n}\sum_{k=1}^{n}\frac{k^{99}}{1}$
So in the integral, what will be the "respect-to-variable?" Would it be:
$\displaystyle \lim_{n\to\infty} \frac{1}{n^{99}} \int_{0}^{1} k^{99} \text{dk}$
$= \displaystyle \lim_{n\to\infty} \frac{1}{n^{99}} \frac{1}{100}$
But that is wrong as shown here: Limit of a summation, using integrals method
Bottonlinequestion: I am confused about how you write an integral from a SUM. Like what is variable the integral is made with respect to?
$$ \sum_{k=1}^{n} \frac{k^{99}}{n^{100}}=\frac1n\sum_{k=1}^{n}\Bigl(\frac{k}{n}\Bigr)^{99}. $$ This is a Riemann sum for the integral $$ \int_0^1 x^{99}\,dx. $$