4
votes
0answers
104 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 ...