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 […]

It is well known that in the case of a finite interval $[0,1]$ with a partition of equal size $1/n$, we have: $$\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)=\int_0^1 f(x)dx$$ I was wondering under which conditions on $f$ this could be extended to the case of the positive real line, i.e which conditions on $f$ would enable us […]

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.

Let $f(x)=x^d$ $(d\in(-1,0))$. We know that $$\sum_{i=1}^n \frac{1}{n}\left(\frac{i}{n}\right)^d\xrightarrow{n\rightarrow\infty}\int_0^1x^d dx=\frac{1}{d+1}.$$ My question is the following: Can we say something about the speed of convergence? Something like $$\left|\int_0^1x^d dx-\sum_{i=1}^n \frac{1}{n}\left(\frac{i}{n}\right)^d\right|\in O(n^d)?$$ I know that the last expression might be wrong. I just wanted to give you an idea what I am looking for. Thanks!

From Spivak’s Calculus, For the theorem: If $f$ is bounded on $[a,b]$, then $f$ is integrable on $[a,b]$ if and only if for every $\varepsilon > 0$ there is a partition $\mathcal{P}$ of $[a,b]$ such that $U( f, \mathcal{P}) – L( f, \mathcal{P}) < \varepsilon.$ Part of the proof is: If $f$ is integrable sup${L(f, […]

Intereting Posts

Computer Assisted proofs apart from the 4 color theorem
Is there a rigorous definition of the term “coordinate system”?
Dirac delta integral with $\delta(\infty) \cdot e^{\infty}$
Simple example of non-arithmetic ring
Eigenvalues of $A+B$
Evaluate an infinite product in a closed form
proving that $SO(n)$ is path connected
Textbook with proof that product of two Gaussian functions is also Gaussian
Interesting Question on Ants
Mollifiers: Asymptotic Convergence vs. Mean Convergence
Axiom of Choice and Cartesian Products
A partition of the unit interval into uncountably many dense uncountable subsets
About the Legendre differential equation
$g^k$ is a primitive element modulo $m$ iff $\gcd (k,\varphi(m))=1$
Commas separating adjectives to describe mathematical objects.