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.

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 $\sum\limits_{k=2}^{n}\dfrac{1}{k}<\log(n)<\sum\limits_{k=1}^{n-1}\dfrac{1}{k}$ 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 […]

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 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 […]

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 […]

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 $x_i$ […]

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 […]

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. 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 […]

Intereting Posts

numbers from $1$ to $2046$
Let $A \subseteq X$ and $f: X \mapsto X$. Prove $f^{-1}(A) = A \iff f(A) \subseteq A \land f^{-1}(A) \subseteq A$
Lagrange method with inequality constraints
Intuitively and Mathematically Understanding the Order of Actions in Permutation GP vs in Dihereal GP
Reference Books on Cryptography
Can we prove that all zeros of entire function cos(x) are real from the Taylor series expansion of cos(x)?
$\sum \cos$ when angles are in arithmetic progression
If $f(z)=(g(z),h(z))$ is continuous then $g$ and $h$ are as well
Counterexample for numerical series
$\mathbb N$ when given the metric $d(m,n)=\frac{1}{m}-\frac{1}{n}$
Determine the centre of a circle
Invalid Argument vs. Contradiction?
is the dual of a finitely generated module finitely generated?
Is a left invertible element of a ring necessarily right invertible?
Sequence of partial sums of e in Q is a Cauchy sequence.