# Riemann Integrals - Evaluating Summations

Can anyone help me solve these please?

I tried following steps given by a teacher:

Defining the partitions in the form of c+k/n Take out 1/n width Creating the function What do we input? The Upper bound? The Geometric average? Define the height of the rectangle in terms of some function of the input Show that the summation is the Riemann sum for the integral and evaluate

However, I haven't been able to solve it after two hours :( Help would be greatly appreciated. Thanks :)

• Please use MathJax to format. – Saad May 5 '20 at 14:24

In fact $$\begin{eqnarray} &&\lim_{n\to\infty}(1^a+2^a+\cdots+n^a)/n^{a+1}\\ &=&\lim_{n\to\infty}\sum_{k=1}^n\frac{k^a}{n^{a+1}}=\lim_{n\to\infty}\frac1n\sum_{k=1}^n(\frac{k}{n})^a\\ &=&\int_0^1x^adx=\frac{1}{a+1},\\ &&\lim_{n\to\infty}((2n+1)^{-1}+(2n+2)^{-1}+\cdots+(3n)^{-1}\\ &=&\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{2n+k}\\ &=&\lim_{n\to\infty}\frac1n\sum_{k=1}^n\frac{1}{2+\frac{k}{n}}\\ &=&\int_0^1\frac{1}{2+x}dx=\ln\frac{3}{2}. \end{eqnarray}$$