2
votes
0answers
56 views

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: ...
1
vote
3answers
133 views

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 ...
0
votes
4answers
138 views
0
votes
2answers
35 views

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 ...
3
votes
1answer
35 views

Asymptotic approximation of a certain sum

During calculations of an expectation of some random variable, I have encountered the following sum: \begin{equation} \sum_{t=2}^{n+1} \frac{t(t-1) \cdot n!}{(n-t+1)!\cdot n^t} \end{equation} I ...
5
votes
1answer
99 views

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 ...
0
votes
0answers
27 views

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: ...
6
votes
2answers
182 views

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 ...
0
votes
0answers
40 views

Approximation for the logarithm of a summatory

I would like to find an approximation for: $$ \log \left(\sum_{i=1}^{N} a_i\exp(-b_i^2)\exp(-c_i^2)\right) $$ with $$ a_i = \frac{1}{\sqrt{(e^2 + e_i^2)(g^2 + g_i^2)}} \\ b_i = \frac{b-d_i}{2(e^2 + ...
4
votes
2answers
171 views

Evaluate $\sum_0^\infty \frac{1}{n^n}$

Courtesy of this xkcd comic I now know that $$ \sum_{n=1}^\infty \frac{1}{n^n} \approx \ln^e(3) $$ Echoing the views of the comic itself, if I ever find myself taking $\ln^e(x)$ then something has ...
1
vote
2answers
77 views

Why sum of sigmoids is a good approximation of softplus function?

According to this paper: Rectified Linear Units Improve Restricted Boltzmann Machines, $\sum_{i=1}^N \sigma(x-i+\tfrac{1}{2}) \approx \log(1+e^x)$ (equation 7) where $\sigma(z) = \frac{1}{1+e^{-x}}$ ...
1
vote
1answer
101 views

Mysterious subleading corrections to sums with internal dependence on limit

Is there a standard method for finding expansions in $N$ of sums like $$S(N)=\sum_{n=0}^N \sqrt{N^2-n^2}$$ beyond the first term? It is easy to compute here that $$S(N)=N^2 \int_0 ^1 \sqrt{1-x^2} ...
1
vote
0answers
37 views

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 ...
2
votes
2answers
168 views

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?
3
votes
1answer
200 views

Asymptotic approximation of sum $\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$

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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
3answers
89 views

Sum of sequence precision

I came up with this answer in stackoverflow. It states a question: ...
2
votes
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 $z = \frac{1}{n-2}$?
0
votes
1answer
222 views

Techniques for bounding a sum

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. ...
2
votes
0answers
140 views

approximation of the sum

I have difficulties to find an approximation formula (or bound from the below) for the following sum: $$ \sum_{k=1}^n\left( \frac{1}{35}\right)^{k-1}(n-k)!\left(k-\frac 32\right)!. $$