# Tagged Questions

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### 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
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### 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: $a=0.1876$, $b=-0.1895$, ...
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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 $$\| f - p_n \|_\infty \leq \frac{1}{(n+1)!} \| f^{(n+1)} \|_\infty ... 1answer 25 views ### Is the Lagrange polynomial integer-valued for points with consecutive integer x-values? What I'm really wondering is, does Lagrange polynomial interpolation have an answer for every question of "what's the next integer in this sequence"? Does it define an infinite integer sequence to ... 0answers 115 views ### Could 4+2+4+2+4+2+\cdots = -1 ? In physics classes, on this StackExchange and even in blogs the sum 1 + 2 + 3 + 4 + \cdots = - \frac{1}{12}  has been under the microscope. Why does 1+2+3+\dots = {-1\over 12}? The ... 6answers 2k views ### What is the pattern to this sequence?$$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 ...
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: ...