0
votes
1answer
16 views

Increasing numbers of interations, patterns

Write expression for e to the power of i with increasing numbers of interations, simplifying wherever possible, comment on patterns discovered throughout the equation. Help would be appreciated
2
votes
2answers
80 views

Approximate formula for the series: $\sum_{k=1}^{+\infty}\dfrac{k^x}{(k!)^x}$

I found that this series: $$S(x)=\sum_{k=1}^{+\infty}\dfrac{k^x}{(k!)^x}$$ can be very well approximated in this way: $$S(x)=\dfrac{1}{\left(a+b\exp(cx)\right)^d}$$ with: $a=0.1876$, $b=-0.1895$, ...
0
votes
1answer
39 views

Why does the interpolation error go to zero if we increase the number of sampling points?

This question is motivated by polynomial interpolation. We know that for $f\in C^{n+1}[a,b]$ and $a=x_0<\dots<x_n=b$ holds $$\| f - p_n \|_\infty \leq \frac{1}{(n+1)!} \| f^{(n+1)} \|_\infty ...
3
votes
1answer
25 views

Is the Lagrange polynomial integer-valued for points with consecutive integer x-values?

What I'm really wondering is, does Lagrange polynomial interpolation have an answer for every question of "what's the next integer in this sequence"? Does it define an infinite integer sequence to ...
4
votes
0answers
115 views

Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The ...
8
votes
6answers
2k views

What is the pattern to this sequence?

$$0, 1, 3, 13, 51, 205$$ More specifically, $$(0,0)\quad(1,1)\quad (2,3)\quad (3,13) \quad(4,51)\quad (5,205)$$ I have tried using the interpolation feature in Grapher.app and Wolfram Alpha, but ...
1
vote
0answers
33 views

Constructing a sequence [duplicate]

Possible Duplicate: Terms of a Sequence Construct a sequence of interpolating values $Y_n \text{ to }f(1 + \sqrt{10})$, where $f(x) = (1 + x^2)^{-1}$ for $-5 \leq x \leq 5$, as follows: ...
4
votes
4answers
192 views

What are some “natural” interpolations of the sequence $\small 0,1,1+2a,1+2a+3a^2,1+2a+3a^2+4a^3,\ldots $?

(This is a spin-off of a recent question here) In fiddling with the answer to that question I came to the set of sequences $\qquad \small \begin{array} {llll} ...
0
votes
2answers
52 views

Interpolating a period-two sequence ($f(x) = f(x-2)$): why $(-1)^x$?

Wolfram|Alpha gives the recurrence equation solution $f(x) = c_2(-1)^x + c_1$. Why is the interpolating function $(-1)^x$? Other functions like $\cos(\pi x)$ (the real part) and even ...