Summation of the series using definite integral:

Question

I am self learning Real Analysis from Elementary Analysis: The Theory of Calculus by Kenneth A. Ross

Today on this website, I learned this new technique to find the summation of series using integration.

Summation of the series using definite integral:

$$\lim \limits_{n\to \infty }\frac{1}{n} \sum \limits^{h(n)}_{r=g(n)}f(\frac{r}{n})=\int \limits^{b}_{a}f(x)dx$$

Where

1.$$\sum \to \int$$

2.$$\frac{r}{n} \to x$$

3.$$\frac{1}{n} \to dx$$

4.$$a=\lim \limits_{n\to \infty }\frac{g(n)}{n}$$

5.$$b=\lim \limits_{n\to \infty }\frac{h(n)}{n}$$

I want to practice this technique by working out some problems on my own. The book I am reading does not deal with this topic.

I request you to recommend me some books where I can find some problems to practice this technique.

This may not be allowed here but if people have some practice problems in mind, I request them to write in comment section.

Answer 1

In particular, if $g(n)=1$ and $h(n)=n$ then this sum is same as Riemann sum over the interval $[0,1]$.

The Book, namely,

Problems in Mathematical analysis III: Integration : W. J. Kaczor and M. T. Nowak

deals some problems related to this!

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Edit: I add three more Books.

1. A course in calculus and real analysis: S.R. Ghorpade and B. V. Limaye

Read section $6.4$ with examples and exercises $37$ and $38$

2. Calculus Vol I: Apostol

Exercise $10.4$: $35$ ( six problems to practice)

3. Calculus : M. Spivak

Section $22$, exercise $9$( six interesting problems to practice)