1
vote
0answers
36 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
1
vote
0answers
42 views

Approximating arccos(a/(a+x)) for the sake of simplfying an integral

I recently tried to evaluate $$\int e^{\beta\arccos(a/(a+x))}dx$$ (everything constant except $x$) and got a complicated answer involving a hypergeometric series with complex arguments. Can anyone ...
1
vote
1answer
92 views

Find $\sin(1/10)$ to within error of $10^{-7}$

The maclaurin series of $\sin(x)$ is $x- x^3/3! + x^5/5! - \cdots + (-1)^n x^{2n+1}/(2n+1)!$. My teacher wants me to use Taylor's inequality theorem on page 607 to solve this problem. I know that ...
0
votes
0answers
21 views

Data estimation based on progression

Given a data-set $x$ and $y$. x | y ------------------ 153,000 | 0.058848 332,641 | 0.36352 506,629 | 0.53 If $x$ being the number of database records ...
4
votes
1answer
212 views

Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$

By integral test, it is easy to see that $$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$ converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$] I am ...
2
votes
3answers
53 views

Series evaluated to $m$ terms, approximating the error

Given a series $\displaystyle\sum_{n=0}^\infty a_n$, how can we bound the error (which I shall denote with $R_n$) when we evaluate it to $m$ terms? $$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$ ...
3
votes
1answer
47 views

Estimate the given sum.

This is the question from "Data Structures and Algorithm Analysis in C" By Mark Weiss. It is the question 1.7. It goes as follows:- Estimate the sum ...
2
votes
2answers
80 views

Error Term in Passing from Summation to Integral

I encountered the following in a paper and do not understand how the error term is being bounded. In what follows, $n$ and $k$ are large integer constants. $$ \sum_{i=0}^{n-1} \ln\left(1 - ...
2
votes
1answer
195 views

Best and most efficient way to numerically compute $e$?

There are many well-known methods for efficiently numerically computing $\pi$, such as Chudnovsky's Method or perhaps Gauss-Legendre's algorithm. I was wondering what the best method for computing $e$ ...
39
votes
10answers
2k views

What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...