# Tagged Questions

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### How to construct a function to map coefficients?

Surely this question is known by many people but I lack of enough maths knowledge so I prefer ask here. I have a triangular matrix that represent coefficients, all of them are rational numbers ...
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### How does one graph $\sum_{x=0}^{n}$ [closed]

How does one graph a summation, like $$\sum_{x=0}^{n} n$$ Can it be like this Because if you take the points from the summation (0,0), (1,1), (2,3), (3,6) you can tell by summations it only works ...
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I have an integral which involves Bessel function as follows: $I=\int_{r=0}^a \int_{\theta=0}^{2\pi}(e^{-jkr\cos(\theta-\phi)}d\theta)rdr$ I have tried with $e^{-jkr\cos(\theta-\phi)}=\sum ... 6answers 221 views ### Fastest way to integrate$\int_0^1 x^{2}\sqrt{x+x^2} \hspace{2mm}dx $This integral looks simple, but it appears that its not so. All Ideas are welcome, no Idea is bad, it may not work in this problem, but may be useful in some other case some other day ! :) 1answer 46 views ### Is an integral without a differential component on a finite number of points just a sum? Is an integral $$\int_{\lbrace 1, 2, 3 \rbrace} f(x)$$ simply the sum $$\sum_{x=1,2,3} f(x)?$$ I ask this question because of the generalization to multiple dimensions of integration by parts ... 1answer 71 views ### Rewrite and approximate the sum as an Integral$\sum_{i=1}^{1000} \sqrt{i}$[closed] This is not an Infinite sum !, how do we change this to an Integral.$ $We normally write an integral as an infinite sum. 2answers 34 views ### If a sequence$\{a_n\}$satisfies the Inequality$a_{n+1} < ka_{n}$, then show that$ \lim\limits_{n \to \infty} a_n =0$where$0< k , a_n< 1$I know one solution. Consider$\sum a_n$Then use ratio test to show that the series converges, hence the sequence. Any other Ideass ! 1answer 50 views ### Finding the limit of an integral Evaluate $$\displaystyle\lim_{j\rightarrow \infty} \displaystyle\int_{0}^{a} \frac{1}{j!} \left(\ln \left(\frac{A}{x}\right)\right)^{j}dx$$ 1answer 46 views ###$\sum_{x=a}^{b-1}\frac{1}{x}$and$\sum_{x=a+1}^b\frac{1}{x}$I have to prove the following relations:$\sum_{x=a}^{b-1}\frac{1}{x}\geq\log b - \log a \sum_{x=a+1}^{b}\frac{1}{x}\leq\log b - \log a $I tried to use the relation that$\int_a^b \frac{1}{x} ...
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If we are given a function, $f(x)$, we can either integrate it or sum it. I'm wondering what integration can do with $f(x)$ that summation can't, and what summation can do that integration can't. ...
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### Can we possibly exchange summation and integration with negative values?

This is an attempt to go further than this answer. Essentially, we have either a summation of an integral: $$\sum_x{ \left( \int{ f(x)dx } \right) } \tag{1}$$ ...or an integral of a summation: ...
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### When can we use substitution for both integrals and summations?

This question is partially inspired by Qiaochu Yuan's answer to "Will moving differentiation from inside, to outside an integral, change the result?". Essentially, I would like to know, if we have: ...
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### Formula for area under the curve

I don't know that the equation that I am going to explain below is correct or not, and this is why I am asking this question. So, I have found out that area under the curve could be found out by ...
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### Application of Residue Theorem and limits

I am trying the following problem from Fisher's Complex Variables book: If $f$ is analytic on a plane except at poles $\gamma_1, \cdots \gamma_N$ and none of them are integers and ...
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### Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx.$$ Here is what I tried (but I do not think this is ...
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### Sum as an integral

Recently I have encountered weird notation that I don't see into. When I have some infinite sum $$\sum_{n=1}^{\infty}f(n)$$ I would rewrite it without thinking to the integral form like this ...
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### Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
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### 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 ...
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### 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 ...
I want to make an integral. I know that Integral and Sum can be exchanged. But if I have the following case? $$\int\left(\sum_{i_1}\sum_{i_2}e^{ix}\right)\sum_{j_1}\sum_{j_2}e^{jx}\,\text dx$$ ...