# Tagged Questions

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### How to find the approximate basic frequency or GCD of a list of numbers?

I could't actually summarize the question in the title, so I'll explain my situation. I want to tell the integer numbers which act as the best approximate basic frequencies of a list of real numbers: ...
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### Approximate summation of the given equation

I have been trying from an hour to approximate the value of $M$ in the equation given below. $$M = \sum\limits_{i=1}^n\left(\sum\limits_{j=1}^n\left(\sqrt{ i^2 + j^2 }\right)\right)$$ One thing I ...
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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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### Evaluating a taylor series around a given point

So I'm having some trouble with the problem: Given that $\ln(x+1)=\sum_{n=1}^{\infty } \frac{(-1)^{n+1}}{n}x^{n}, -1<x\leq 1$, find the Taylor series of ln(x) around 3. For which x is this series ...
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### Asymptotic approximation of a certain sum

During calculations of an expectation of some random variable, I have encountered the following sum: $$\sum_{t=2}^{n+1} \frac{t(t-1) \cdot n!}{(n-t+1)!\cdot n^t}$$ I ...
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### How to show this: $\sum_{k=2}^{n}\frac{\ln{k}}{k^2}\approx \ln{n}\cdot\left(\zeta_{n}{(2)}-\frac{\pi^2}{6}\right)+C$

Show that: $$\sum_{k=2}^{n}\dfrac{\ln{k}}{k^2}\approx \ln{n}\cdot\left(\zeta_{n}{(2)}-\dfrac{\pi^2}{6}\right)+C,n\to\infty$$ where $$\zeta_{n}{(k)}=\sum_{j=1}^{n}\dfrac{1}{j^k}$$and $C$ is ...
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### Approximation of sum with binomial summands

I am new here, so hopefully my question will be understood correctly. I have a function (originating from expected untility theory in economics) that looks the following: ...
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### Find the integral part of $\sum_{i=2}^{10000}\frac1{\sqrt{i}}$

$$A = \frac1{\sqrt{2}}+\frac1{\sqrt{3}}+\cdots+\frac{1}{\sqrt{10000}}$$ Find $\lfloor A\rfloor$ where $\lfloor x\rfloor$ is the greatest integer less than, or equal to $x$ I got stuck on this, so ...
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### Approximating sums

I got a general question, that is motivated by a recent problem. So let me first describe the problem and then add the general part: I got a rather simple (using only basic elements) equation, which ...
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### Sums of central binomial coefficients

Are there closed forms for $$\sum^n_{i=0} \binom{2i}{i}$$ and $$\sum^n_{i=0} \binom{2i}{i}^2$$? Also, how can these sums be approximated?
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I need to find the asymptotic approximation of this sum $$\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$$ Can you please share a link to theory or hint how it can be solved? Here is my attempt $n ... 1answer 97 views ### Which form of Euler-Maclaurin formula to use? This question may be rather elementary, but I am sort of confused about various forms of the Euler-Maclaurin summation formula and their use. For instance, let us suppose that we want to approximate ... 3answers 55 views ### Estimating$\sum_{k=1}^N a_kb_k$given the means$\bar a_k,\bar b_k$and determining the error I need to calculate the following expression: $$\sum_{k=1}^N a_k b_k$$ I know the average values of$a_k$, defined as$\overline {a_k} = {\sum_{k=1}^N a_k \over N } $and$b_k$, defined as ... 3answers 89 views ### Sum of sequence precision I came up with this answer in stackoverflow. It states a question: ... 0answers 97 views ### Good upper bound on a binomial sum What is a good upper bound on the following binomial sum: $$\sum_{i,j< \frac{m}{n}} {m \choose i}{m-i \choose j} z^{i+j}$$ where$z = \frac{1}{n-2}\$?
I have a very messy function. It consists sums four levels deep, and the inner-most term is itself quite messy. $$\sum \sum \sum \sum (\mbox{stuff})$$ I can't find a closed form for this function. ...
I have difficulties to find an approximation formula (or bound from the below) for the following sum: $$\sum_{k=1}^n\left( \frac{1}{35}\right)^{k-1}(n-k)!\left(k-\frac 32\right)!.$$