# Tagged Questions

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### What's the purpose of this formula?

Just found this image on the web: Can anyone explain what's the meaning (if any) of this formula? (I did a Google image search but found no answer)
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### If $\lim\limits_{x \to \infty} f(x) = 1$, can we have function $f(x)$, such that $\int_0^{\infty}f(x)dx$ converges

I know the Initiative answer, can anyone give a neat answer based on solid reasoning EDIT : $f(x)$ is continuous
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### How large should $a$ be so that $\int_a^{\infty} \frac{dx}{1+x^2} < \frac{1}{1000}$

I want to solve this without using calculator.
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### What is this sequence of polynomials?

NovaDenizen says the polynomial sequence i wanted to know about has these two recurrence relations (1) $p_n(x+1) = \sum_{i=0}^{n} (x+1)^{n-i}p_i(x)$ (2) $p_{n+1}(x) = \sum_{i=1}^{x} ip_n(i)$ == i ...
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### Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
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### Is this proof correct?

I was working on a summation problem, and I thought of a way to solve it. This is a proof of a generalisation of that method, is it correct, can it be improved? We will describe the conditions ...
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### compare summation and integral

If we know $f(x)$ is monotonic decreasing on the interval $a \leq x < \infty$, could we obtain following relation formally? \int_{m}^{n+1} f(x) dx \leq \sum_{j=m}^n f(j) \leq ...
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### Prove a inequality about integral and summation

If $f(x)$ is monotonic increasing on the interval $a\leq x < \infty$, could we prove following inequality formally? f(a+k) \leq \int_{a+k}^{a+k+1} f(t) dt \leq f(a+k+1) ...
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### Prove that $\lim_{n\to\infty} H_n/n = 0$ ($H_n$ is the $n$-th harmonic number) using certain techniques

I can't seem to use certain methods such as $\varepsilon$-N, L'Hôspital's Rule, Riemann Sums, Integral Test and Divergence Test Contrapositive or Euler's Integral Representation to prove that ...
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### Evaluating the following sum

I have no idea how to solve evaluate this integral: $$\lim_{n\to\infty} \frac{1^a + 2^a + \cdots + n^a}{n^{1+a}}, a > -1$$ I want to set this up as some sort of integration since it is a ...
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### Proofs of $\sum_{n=1}^\infty \frac{1}{n^n}=\int_0^1 \frac{1}{x^x}dx$ [duplicate]

Prove that $\sum_{n=1}^\infty \frac{1}{n^n}=\int_0^1 \frac{1}{x^x}dx$. This comes from this question Evaluate $\sum_0^\infty \frac{1}{n^n}$.
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### How to solve summation problem?

I want to evaluate " sum f(x) g'(x)" the limits of sum goes from 0---> infinity. I know how to solve this problem using integration by substitution method. I also know that using Riemann sum we can ...
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### How do I express the integral of $\tan^{2n}(x)$ using sigma notation?

How do I express the integral of $\tan^{2n}(x)$ with respect to $x$ using sigma notation? $$\int \tan^{2n}(x)\, dx = \text{???}$$
This is what I want to show: $\sum_{i=0}^n 1/i \leq c \log n$ for all $n>N$ My current approach was this: \sum_{i=1}^n 1/i = ( \int \sum_{i=1}^n 1/i )' = ( \sum_{i=1}^n \int 1/i )' = ( ... 1answer 138 views ### Find an expression for the area under the graph of f(x) as a limit? f(X) = 2x/(x^2 +1), 1 <= x <= 3 Basically, I need to find an expression for the area under the graph within these intervals for the function as a limit. I understand the concept of the area ... 0answers 42 views ### calculating sum of a limit of integral I am trying to calculate the following expression $$\sum_{m=0}^{\infty} \frac{1}{m!} \lim_{n \to \infty} \int_{\{(x,y):2x^2+y^2<n^2 \}}\left( 1 - \frac{2x^2+y^2}{n^2}\right)^{n^2} x^{2m}dx~dy ... 2answers 269 views ### maybe this sum have approximation \sum_{k=0}^{n}\binom{n}{k}^3\approx\frac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty prove or disprove this$$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$this problem is from when Find this limit ... 2answers 170 views ### Difference between summation and integration It is well known that if a series \sum\limits_{k= 0}^\infty a_k converges, then a_k \to 0. However, this is not true for integrals. What makes them different? Is it simply that they are ... 1answer 44 views ### Integrating over sums. I want to make an integral. I know that Integral and Sum can be exchanged. But if I have the following case?$$ \int\left(\sum_{i_1}\sum_{i_2}e^{ix}\right)\sum_{j_1}\sum_{j_2}e^{jx}\,\text dx $$... 2answers 101 views ### Finding the complex fourier series of the function x^2sin(x) in the interval [{-\pi}, \pi]? This forms part of a project I am doing and I wish to see how well complex fourier series approximates a smooth curve such as this one. After tedious integration by parts, I have attained an answer ... 1answer 29 views ### integral alternative of \sum f(p) from (1<\text{all primes}\leq n) to (\text{maximum prime}<n) I have a naive question that if someone could find the integral alternative of$$\sum_{\substack{2\le p\le n\\p\text{ is prime}}} f(p)$$where f(p) is a non-decreasing monotonic function. 2answers 107 views ### Fourier Series Coefficient of a given signal$$ {\rm x}\left(t\right) = \sum_{k = -\infty}^{\infty}\left[\delta\left(t-\dfrac{k}{3}\right) + \delta\left(t-\dfrac{2k}{3}\right)\right] I need to find the Fourier series coefficient of x(t). I ... 1answer 43 views ### When can we interchange Fourier transform and countable sum? When does \mathcal{F}\left ( \sum_{n=1}^{\infty} f_n (x)\right ) = \sum_{n=1}^{\infty} \mathcal{F}(f_n(x)) where \mathcal{F} the Fourier transform operator. 0answers 134 views ### Is there a closed form for this sum? While generalizing the previous result, I conjectured that the series expansion of \begin{align*} \int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y ... 2answers 77 views ### How to interpret this formula from dsp? [closed] Can you describe me what this formula does and what result in the end?\delta(x)=\begin{cases} 0, & \text{ifx \neq 0$} \\ 1, & \text{if$x = 0$} \\ \end{cases}$$... 0answers 35 views ### Show that \lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}= \frac{\pi}{4} [duplicate] Show that:$$ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}= \frac{\pi}{4}$$My idea: I thought that this could be rewritten as an integral, then use trig substitution (perhaps tangent) and then ... 2answers 173 views ### \int_0^{\frac{\pi}{2}}x\cot(x)dx and \lim_{m \rightarrow \infty}\log\left( e^{2m}\left(\frac{(2m-1)!!}{(2m+1)^m}\right)^2\right). I'm trying to evaluate the integral, but in doing so have stumbled upon the limit, which I don't know whether it exists, and if so how to resolve it (and whether I've derived the relationship between ... 2answers 94 views ### Prove that \lim\limits_{n\rightarrow\infty} \displaystyle\sum\nolimits_{k=1}^n\frac{n}{n^2+k^2x^2} =\frac{\tan^{-1}(x)}{x} [closed] Prove that$$\lim\limits_{n \to + \infty} \left( \sum\limits_{k=1}^n \frac{n}{n^2+k^2x^2} \right) =\frac{\tan^{-1}(x)}{x}.$$2answers 43 views ### Find the antiderivative of a function with a finite series and factorials If n\in\mathbb{N},s\leq n, I know that$$ \int_0^1 t^s(1-t)^{n-s-1}dt=\frac{s!(n-s-1)!}{n!}. $$I would like to find a similar formula: is there a function f(t) such that$$ \int_0^1 f(t) ... 1answer 50 views ### How$\sum_{r=m}^{\infty}\frac{e^{-\lambda}\lambda^r}{r!}=\int_{0}^{\lambda}\frac{e^{-u}u^{m-1}}{(m-1)!}du$$$P(X\geq m)=\sum_{r=m}^{\infty}\frac{e^{-\lambda}\lambda^r}{r!};m=0,1,...$$ Show that for any$m=1,2,...$$$P(X\geq m)=\int_{0}^{\lambda}\frac{e^{-u}u^{m-1}}{(m-1)!}du$$ I couldn't derive it also ... 2answers 109 views ### Finding non-trivial functions$f(x)$such that$\sum_{n=1}^{\infty}f(n)=\int_{0}^1f(x)dx\$
It is known that Atle Selberg found the following expression when he was 14 years old : $$\sum_{n=1}^{\infty}n^{-n}=\int_{0}^1x^{-x}dx.$$ Then, here is my question. Question : Find the other ...
Prove$$\sum_{k=0}^x \binom{n}{k}p^{k}(1-p)^{n-k} =(n-x)\binom{n}{x}\int_{0}^{1-p}t^{n-x-1}(1-t)^{x}dt.$$ Can someone show me the steps please? Here is the hint my book gave me: "Integrate by parts ...