Intereting Posts

Relating $\operatorname{lcm}$ and $\gcd$
How prove this $|A||M|=A_{11}A_{nn}-A_{1n}A_{n1}$
Differentiating an integral using dominated convergence
Reference request: certain special LFTs
Is it possible to describe the Collatz function in one formula?
Does Bounded Covergence Theorem hold for Riemann integral?
Can forcing push the continuum above a weakly inacessible cardinal?
evaluating sum of limits
Express integral over curve as integral over unit ball
Counting squarefree numbers which have $k$ prime factors?
Why General Leibniz rule and Newton's Binomial are so similar?
Why does $\frac{dq}{dt}$ not depend on $q$? Why does the calculus of variations work?
$(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$
Limit of this recursive sequence and convergence
Example of a compact set that isn't the spectrum of an operator

So how to approach this one? $\frac1n\sum g(\frac{r}{n}) $ . How to convert in this form? As I can see r and n will have different powers.

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- How to choose a proper contour for a contour integral?
- Proving that an integral is differentiable
- How to integrate $\int_{0}^{\pi} \frac{1}{a-b \cos(x)} dx$ with calculus tools?
- How to integrate $\int \frac{dx}{\sqrt{ax^2-b}}$

- Using Spherical coordinates find the volume:
- Proving that $\int_0^{\pi/2} (\sin (\tan (x))+\cot (x) \cos (\tan (x))-\cot (x))\cot (x) \, dx=\frac{\pi(e-2)}{2e}$
- Tricky contour integral resulting from the integration of $\sin ax / (x^2+b^2)$ over the positive halfline
- Evaluate the integral $\int_{0}^{+\infty}\frac{\arctan \pi x-\arctan x}{x}dx$
- How find this limit $\lim \limits_{x\to+\infty}e^{-x}\left(1+\frac{1}{x}\right)^{x^2}$
- What is the significance of the three nonzero requirements in the $\varepsilon-\delta$ definition of the limit?
- Why can't I use the disk method to compute surface area?
- closed form of $\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^n}$
- Evaluate $\int _0^{\infty }\frac{e^{-t^2}-e^{-4t^2}}{t^2}dt$
- three term asymptotic solution as $\epsilon \to 0$

Since

$$\frac{1}{\sqrt{n^2 + n}} + \frac{1}{\sqrt{n^2 + n}} + \cdots + \frac{1}{\sqrt{n^2 + n}} < \frac{1}{\sqrt{n^2 + 1}} + \cdots + \frac{1}{\sqrt{n^2 + n}}$$

and

$$\frac{1}{\sqrt{n^2 + 1}} + \cdots + \frac{1}{\sqrt{n^2 + n}} < \frac{1}{\sqrt{n^2 + 1}} + \cdots + \frac{1}{\sqrt{n^2 + 1}},$$

we have

$$\frac{n}{\sqrt{n^2 + n}} < \frac{1}{\sqrt{n^2 + 1}} + \cdots + \frac{1}{\sqrt{n^2 + n}} < \frac{n}{\sqrt{n^2 + 1}}.$$

As $$\frac{n}{\sqrt{n^2 + n}} = \frac{1}{\sqrt{1 + \frac{1}{n}}} \to 1$$

and

$$\frac{n}{\sqrt{n^2 + 1}} = \frac{1}{\sqrt{1 + \frac{1}{n^2}}} \to 1$$

by the squeeze theorem,

$$\lim_{n\to\infty} \left(\frac{1}{\sqrt{n^2 + 1}} + \cdots + \frac{1}{\sqrt{n^2 + n}}\right) = 1.$$

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