# Questions tagged [riemann-sum]

This tag is for questions about Riemann sums and Darboux sums.

102 questions
1k views

322 views

### Using right-hand Riemann sum to evaluate the limit of $\frac{n}{n^2+1}+ \cdots+\frac{n}{n^2+n^2}$

I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, ...
5k views

### Is the indicator function of the rationals Riemann integrable?

$f(x) = \begin{cases} 1 & x\in\Bbb Q \\[2ex] 0 & x\notin\Bbb Q \end{cases}$ Is this function Riemann integrable on $[0,1]$? Since rational and irrational numbers are dense on $[0,1]$, no ...
4k views

### General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
479 views

471 views

### Convergence of a sequence (possibly Riemann sum)

Let $a_1, a_2, a_3, . . . , a_n$ be the sequence defined by $$a_n = 2\sqrt{n}-\sum_{k=1}^{n}\frac{1}{\sqrt{k}} = 2\sqrt{n} - \frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-...-\frac{1}{\sqrt{n}}$$ show ...
179 views

### limit $\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$

How do I evaluate this? $$\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$$ I got concerned for that, I've tried make it integral for Riemann but it still undone.
133 views

### Riemann Sums proving function identically zero

Let $f$ be continuous on $[a,b]$. Suppose that $f(x)\geqslant 0$ for every $x \in [a,b]$ and that $$\int_a^b f(x)\mathsf dx = 0.$$ Prove that $f$ is identically zero on $[a,b]$. So I know to use ...
3k views

### Sum of divergent series

I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the ...
479 views

### How to integrate $xe^x$ without using antiderivatives or integration by parts.

Yesterday, I sat for my Real Analysis II paper. There I found a question asking to integrate $\displaystyle\int_0^1 xe^x \, dx$ without using antiderivatives and integrating by parts. I tried it by ...
507 views

### Is there any reason to expect the Riemann sum over $[a,b]$ to converge to the definite integral $\int_{a}^{b} f(x) \, dx$?

When learning the definite integral 'rigorously', most first courses seem to follow the steps below. Sketch the function over $[a,b]$ Construct arbitrary left and right function value partitions, ...
2k views

### Let $f(x)=x$ for $x\in[0,1]$ rational and $f(x)=0$ for $x\in[0,1]$ irrational. Prove that $f$ is not Riemann integrable on $[0,1]$.

Let $f:[0,1]\to\mathbb{R}$ where $f(x)=x$ for $x\in[0,1]$ rational and $f(x)=0$ for $x\in[0,1]$ irrational. Prove that $f$ is not Riemann integrable on $[0,1]$. I have a vague idea of what I'm ...
2k views

I am trying to prove the Lebesgue Criterion for Riemann Integrability without using the concept of oscillation. The Lebesgue Criterion for Riemann Integrability states that if $f: [a,b] \to \mathbb{... 2answers 245 views ### Show that$\int_{a}^{c}f(x)dx = 0$for all$c\in [a,b]$if and only if$f(x) = 0$for all$x\in [a,b]$. Exercise: Suppose that$a<b$and that$f:[a,b]\rightarrow R$is continuous. Show that$\int_{a}^{c}f(x)dx = 0$for all$c\in [a,b]$if and only if$f(x) = 0$for all$x\in [a,b]$. attempt of ... 2answers 554 views ###$\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}$using Riemann sums? [closed] How to find the integral $$\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}=1$$ using Riemann sums? 1answer 2k views ### Riemann Sum Approximations: When are trapezoids more accurate than the middle sum? We can approximate a definite integral,$\int_a^b f(x)dx$, using a variety of Riemann sums. If$T_n$and$M_n$are the nth sums using the trapezoid and midpoint (middle) sum methods and if the second ... 3answers 2k views ### Proving Riemann Sums via Analysis Exercise$\bf 5.1.7$: Suppose$f:[a,b]\to\Bbb R$is Riemann integrable. Let$\epsilon\gt0$be given. Then show that there exists a partition$P=\{x_0,x_1,\ldots,x_n\}$such that if we pick any set of ... 1answer 671 views ### Prove Lipschitz function$f$with constant$K$is integrable on$[0, 1].$We suppose that$f : [0, 1] \rightarrow \mathbb{R}$is a Lipschitz function with constant$K$. We want to show that$f$is integrable on$[0, 1].$I've been trying to use the Darboux criterion of ... 1answer 126 views ### Sum of$\sum_{k=1}^n\sin{k\theta}$[duplicate] I have to calculate the sum of this series $$\sum_{k=1}^n\sin{k\theta}$$ I tried solving it like this $$\sum_{k=1}^n\sin{k\theta} = \operatorname{Im}\sum_{k=1}^ne^{(i\theta)k}$$ I recognized it as ... 2answers 98 views ### To evaluate the limits$\lim\limits_{n \to \infty} \{\frac{1}{1+n^3}+\frac{4}{8+n^3}+\ldots +\frac{n^2}{n^3+n^3}\}$To me it seems like that we need to manipulate the given sum into the Riemann sum of some function. First writing in the standard summation form; $$\{\frac{1}{1+n^3}+\frac{4}{8+n^3}+\ldots +\frac{n^2}... 2answers 150 views ### Given f(x), a continuous function on [0, 1] st f(x)≥0 for all x∈[0, 1], show that if \int_0^1 f(x)dx=0 then f(x) = 0 for all x ∈ [0, 1] [duplicate] Given f(x), a continuous function on the interval [0, 1] such that f(x) ≥ 0 for all x ∈ [0, 1], show that if \int_0^1 f(x)dx = 0 then f(x) = 0 for all x ∈ [0, 1]. Is this true if f(x)... 1answer 88 views ### Working with square roots I'm asking for methods on working with and dealing with square roots in algebra. Numerous times square roots have been the bane of my existence and are usually difficult to get around. E.g. the ... 1answer 128 views ### Riemann Lower Sum by definition [closed] I have asked a similar question before, and didn't get the answer I was looking for, so i'll try to be more clear here. If necessary here is the link: Riemann sums upper and lower sums question Okay ... 2answers 531 views ### Does this sum of prime numbers converge? \newcommand{\P}{\operatorname{P}}I'm wondering if this sum of prime numbers converges and how can I estimate the value of convergence.$$\sum_{k=1}^\infty \frac{\P[k+1]-2\P[k+2]+\P[k+3]}{\P[k]-\P[... 7answers 1k views ### Finding limits using definite integrals$\lim_{n\to\infty}\sum^n_{k=1}\frac{k^{4}}{n^{5}}$Find the limit of$\displaystyle\lim_{n\to\infty}\frac 1 {n^5}(1^4+2^4...+n^4)$using definite integrals. It's equal to:$\displaystyle\lim_{n\to\infty} \sum^n_{i=1}\frac 1 i$but now I'm not sure ... 1answer 4k views ### To prove the equivalence definition of Riemann integral. I have some trouble with the Riemann integral, specifically, the definition of it in an article on wikipedia. We say that the Riemann integral of$f$equals s if the following condition holds: For a ... 2answers 628 views ### Riemann sum on infinite interval It is well known that in the case of a finite interval$[0,1]$with a partition of equal size$1/n$, we have: $$\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)=\int_0^... 1answer 211 views ### How to calculate the limit \lim _{n\to \infty }\sum _{k=1}^n\sqrt{n^4+k}\sin(\frac{2k\pi }{n})? Is it a Riemann sum? I just came across this limit and I suppose it can be computed using a Riemann sum but I can't get it right.$$\lim _{n\to \infty }\sum _{k=1}^n\sqrt{n^4+k}\sin\left(\frac{2k\pi }{n}\right)$$Any ... 1answer 166 views ### The limit of a sum I'm trying to find out this limit$$\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^n e^{-\Large\frac{k\pi\sqrt{2}}{4n}}\left(\tan\frac{k\pi}{4n}\right)^2 =?$$My try: I know I have to transform the ... 1answer 3k views ### integration of 1/x as a riemann sum To integrate x^\alpha when \alpha\neq1 we subdivide the interval [a,b] by the point of geometric progression:$$a, aq, aq^2, \ldots, aq^{n-1}, aq^n=b$$where q=\sqrt[n]{b/a}. We then only need ... 4answers 196 views ### Evaluation of \lim_{n\rightarrow \infty}\sum_{k=1}^n\sin \left(\frac{n}{n^2+k^2}\right) Evaluation of \displaystyle \lim_{n\rightarrow \infty}\sin \left(\frac{n}{n^2+1}\right)+\sin \left(\frac{n}{n^2+2^2}\right)+\cdots+\sin \left(\frac{n}{n^2+n^2}\right) \bf{My Try::} We can write ... 2answers 366 views ### Use Darboux sums to calculate the area of \sqrt x in [0,1] I know that$$\Delta x=\frac{b-a}{n}=\frac{1-0}{n}=\frac{1}{n}$$So$$\overline{D}=\displaystyle\lim_{n\to\infty}\displaystyle\sum_{i=1}^n\sup f(x_i)\Delta x=\displaystyle\lim_{n\to\infty}\... 2answers 456 views ### Evaluate$ \lim_{h \to 0} h\sum_{n=0}^{\infty} e^{-n^2h^2}$[closed] Evaluate $$\lim_{h \to 0} h\sum_{n=0}^{\infty} e^{-n^2h^2}$$ I think it is somehow related to Riemann Sums, but I'm not sure. Please help. 2answers 119 views ### Verifying Riemann Sum $$a_n = \frac{1}{n^2} \sum _{k=1}^n \left( \sqrt{\left(n+k\right)^2+n+k}\, \right)$$ $$\lim_{n\to \infty} a_n =\:?$$ I solved it like this and I would want to know it this makes sense and if not why.... 1answer 263 views ### Question on Riemann sums Question is : What I have read in Riemann sum definition is something like $$\sum_{i=1}^n f(y) .|x_i-x_{i-1}|$$ So, at first sight i am afraid this is not even related to Riemann integration of$f$... 3answers 663 views ### 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 ... 3answers 1k views ### proving$\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$when f is continous on [0,1] $$\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$$ when f is continuous on$[0,1]$I know it can be proved using bounded convergence theorem but, I wanna know proof using only basic ... 2answers 771 views ### Understanding the definition of the Riemann Integral In our real analysis course, we have been introduced to the idea of a Riemann Integral. I understand the intuitive concept of the Riemann Integral, but I don't quite understand the definitions of the ... 2answers 1k views ### Prove that If$f$is integrable on$[a,b]$and$[b,c]$then$f$is integrable on$[a,c]$If$f$is integrable on$[a,b]$and$[b,c]$then$f$is integrable on$[a,c]$and $$\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx .$$ Let's define three partitions$P,Q,R$on$[a,c]$,$[a,b]...
Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Suppose that there is a sequence of partitions $\{P_n\}_{n=1}^\infty$ with mesh tending to $0$, $P_n=\{a=t_0^n<t_1^n<\ldots<t_{r_n}^n=b\}$, ...
### Show $\sin{\frac{1}{x}}$ for $x\not= 0$ and $f(0)=0$ is integrable on $[-1,1]$
Show $\sin{\frac{1}{x}}$ for $x\not= 0$ and $f(0)=0$ is integrable on $[-1,1]$. I guess my strategy for solving this is to use the following theorem: Let $f$ be a function defined on $[a,b]$. If \$a&...