Articles of riemann sum

Changing the order of integration (for Lebesgue-Stieltjes integral and Riemann integral)

Do the Lebesgue-Stieltjes integral and the Riemann integral have the same rules about the change of order of integration? I mean I know how to deal with Riemann integral, but I’m not sure if I can simply apply the same rules to the Lebesgue-Stieltjes. Thanks. It’s double integration in $\mathbb{R}^2$, by the way.

How to prove Left Riemann Sum is underestimate and Right Riemann sum is overestimate?

Let $A$ be the exact area over $[a,b]$ under $y=f(x)$. If $f(x) \geq 0$ (positive), and increasing, then $\forall x \in [a,b]$, Left Riemann Sum $\leq$ A $\leq$ Right Riemann Sum. How do I prove this? I don’t know where to start

How to prove $\sum\limits_{k=2}^{n}\dfrac{1}{k}<\log(n)<\sum\limits_{k=1}^{n-1}\dfrac{1}{k}$

How to prove $\sum\limits_{k=2}^{n}\dfrac{1}{k}<\log(n)<\sum\limits_{k=1}^{n-1}\dfrac{1}{k}$ It is clear if i consider the area under $f(x)=\dfrac{1}{x})$ from $1$ to $n$ end divide the interval $[1,n]$ into subintervals of length $1$, since $1/x$ is strictly decreasing LHS of the inequality takes the minimum value and RHS the maximum, so $\log(n)$ is always between them. But I want to […]

Conversion of Riemann Sum to Integral

I was trying to convert a Riemann Sum to an integral however I got the parameters wrong according to the answers from the textbook. Now the Riemann Sum involves a $\ln$ as the function as is as follows: $$\sum_{i=1}^n {2\over n} \ln ( 1 + {2i\over n})$$ I assumed that the $\operatorname{dx}$ would be equal […]

If the left Riemann sum of a function converges, is the function integrable?

If the left Riemann sum of a function over uniform partition converges, is the function integrable? To put the question more precisely, let me borrow a few definitions first. Pardon my use of potentially non-canon definitions of convergence. Given a bounded function $f:\left[a,b\right]\to\mathbb{R}$, A partition $P$ is a set $\{x_i\}_{i=0}^{n}\subset\left[a,b\right]$ satisfying $a=x_0\leq x_1\leq\cdots\leq x_n=b$. The […]

Convert to Riemann Sum

The following limit has to be converted to Riemann Sum. $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{2N}{N^{2}+n^{2}}\right)$$=$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{2}{1+n^{2}/N^{2}}\right).1/N$. Now, replacing $1/N$ by $dx$, $n^{2}/N^{2}$ by $x^{2}$ and summation by integral, we have $$\lim_{N→∞}1/N\sum^{N}_{n=-N}\left(\frac{2}{1+n^{2}/N^{2}}\right)= \lim_{N→∞}[1-(-1)]/N\sum^{N}_{n=-N}\left(\frac{1}{1+n^{2}/N^{2}}\right)$$ $$=2\lim_{N→∞}[1-(-1)]/N\sum^{N}_{n=0}\left(\frac{1}{1+n^{2}/N^{2}}\right)=?$$ I feel that I am very close to the final answer which is $2\int_{-1}^{1} dt/(1+t^{2})$. But I am stuck after this step. please complete […]

integration of 1/x as a riemann sum

To integrate $x^\alpha$ when $\alpha\neq1$ we subdivide the interval [a,b] by the point of geometric progression: $$a, aq, aq^2, \ldots, aq^{n-1}, aq^n=b$$ where $q=\sqrt[n]{b/a}$. We then only need to evaluate the sum of geometric series. Given the points of division $x_i=aq^i$ the length of the ith cell is given by: $$\Delta x_i=aq^i-aq^{i-1}=aq^i(q-1)/q$$ The largest $x_i$ […]

Proving inequalities about integral approximation

We can state that, with $n$ integer, $$\int_1^n \log x \ \mathrm{dx} \leq \sum_{m = 1}^n \log m$$ because the second is the area of $n$ rectangles with unity base, while the first is “just” the area under the function. 1) How can it analitically or geometrically be proved? 2) Can this be stated in […]

Calculate an integral with Riemann sum

We know that Riemann sum gives us the following formula for a function $f\in C^1$: $$\lim_{n\to \infty}\frac 1n\sum_{k=0}^n f\left(\frac kn\right)=\int_0^1f.$$ I am looking for an example where the exact calculation of $\int f$ would be interesting with a Riemann sum. We usually use integrals to calculate a Riemann sum, but I am interesting in the […]

Find $\lim_{n \to \infty} n^2 \sum^n_{k=1} {1 \over {(n^2+k^2)^2}}$ using Riemann sums

Find $$\lim_{n \to \infty} n^2 \sum^n_{k=1} {1 \over {(n^2+k^2)^2}}$$ using Riemann sums. I got $$\lim_{n \to \infty} n^2 \sum^n_{k=1} {1 \over {(n^2+k^2)^2}} = \lim_{n \to \infty} \sum^n_{k=1} {1 \over {n^2}} {1 \over {(1+({k \over n})^2)^2}} $$ Now, this is not the classic $\lim_{n \to \infty} \sum^n_{k=1} {1 \over {n}} {1 \over {1+({k \over n})^2}}$ that […]