# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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### Closed form of coefficients of a finite field polynomial

I want to find a valid polynomial for a finite field $\mathbb{Z}_p[x]_{f(x)}$ with $d=deg(f(x))$. For this definition to hold, it can be deduced that $p$ must be prime and the polynomial $f(x)$ ...
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### Series with Hermite polynomials (Mehler Formula)

After a long calculation, I end up with this series: $$\sum_{n = 0}^{+\infty} \frac{1}{2^n n! (k^2 - 4(2n+1))} H_n(x) H_n(y),$$ where $H_n(x)$ is the physicist's Hermite polynomial and k is a real ...
• 115
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### Fisher information for Poisson distribution

The context for the question is this paper. I am trying to understand how to get from Eq. (5) to Eq. (7). For simplicity I will only consider 1 dimension, whereas the equations in the paper are ...
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### Calculator For Mathematical Sequence [closed]

I'm wanting to calculate a mathematical sequence of adding (54.5782*0.99997105^n) (in which n increases by 1 each time) (until n reaches 436,422). That looks like ...
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### Does $\sum^{\infty}_{n=1} \exp{\left(\frac{-n^\epsilon}{2\log_2n}\right)}$ converge?

I'm working through the proof for the longest run of heads in a Bernoulli process and I'm having some trouble with the infinite series in the title. Let $n$ be the total number of tosses in the ...
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### A confusion about evaluating a sequential limit using the definition of the Riemann integral.

So the question is as below: $$\lim_{n\to +\infty} \left(\frac{1}{n^2+n+1}+\frac{2}{n^2+n+2}+\frac{3}{n^2+n+3}+\cdots +\frac{n}{n^2+n+n}\right)$$ And my strategy to tackle this is to make use of the ...
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### Proving $\sum\limits_{k\ge0}\frac1{2k+2}\left[\frac1{2^{2k}(1-2k)}\binom{2k}k\right]^2=\frac{16}{9\pi}$
Whilst looking at Jack's answer to this question, he claims that $$\sum_{k\geq 0}\frac{1}{2k+2}\left[\frac{1}{2^{2k}(1-2k)}\binom{2k}{k}\right]^2=\frac{16}{9\pi}$$ and, as suggested by OP, this result ...