0
votes
2answers
51 views

summation of ceil and floor function

I need a closed solution or a faster algorithm for calculating $$ \sum_{k=1}^{n-1} \left\lceil \frac{n}{k}-1 \right\rceil $$ and $$ \sum_{k=1}^{n-1} \left\lfloor \frac{n}{k} \right\rfloor $$ where $ ...
0
votes
0answers
31 views

Is there a way to expand Re(Li(a^z)) in series?

I'm searching a way to expand $ f(z) = Re(Li(a^z)), a \in R, z \in C $ in series. The computer-friendly, quickly convergent series is a huge plus. For being 'computer-friendly' I mean a relatively ...
3
votes
3answers
72 views

A good way to find $a_{50000}$ where $a_n$ is a number in the form of $2^j\cdot 3^k$

Letting $A=\{2^j\cdot 3^k| j,k \ \text{are non-negative integers} \}$, let us define $a_n$ as the $n$-th element of $A$ in ascending order. We can see $$a_1=1, ...
0
votes
0answers
107 views

Computation mathematics, sequences and roots

a) For $n=1,2,3..,$ let $I_n = \int_0^1 \frac{x^{n-1}}{2-x} dx$ Writing $x^n = x^{n-1}(2-(2-x))$, show that this sequence of numbers satisfies the recurrence relation: $I_{n+1} = 2I_n - ...
4
votes
6answers
3k views

Fastest Square Root Algorithm

What is the fastest algorithm for finding the square root of a number? I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
6
votes
2answers
377 views

Accelerating Convergence of a Sequence

Suppose I had a monotonically increasing sequence $\{d_{n}\}$ which is also bounded above. The $d_{n}$'s satisfy a given recurrence, however computationally they tend very slowly to the limit. What ...
5
votes
3answers
399 views

Calculate $\pi$ in an arbitrary base, to arbitrary precision

I need to calculate $\pi$ -- in base: 4, 12, 32, and 128 -- to an arbitrary number of digits. (It's for an artist friend). I remember Taylor series and I've found miscellaneous "BBP" formulas, but so ...
10
votes
2answers
868 views

How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction?

Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions. Description of the Bauer-Muir transformation ...