# Tagged Questions

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### Zeta function and probability

I know that $\zeta(n) = \displaystyle\sum_{k=1}^\infty \frac{1}{k^n}$ (Where $\zeta(n)$ is the Riemann zeta function) But the reciprocal of $\zeta(n)$ for $n$ a positive integer is equal to the ...
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### Proof for $\displaystyle\sum_{k=1}^n k^a$ equaling a sum of fractions

I know $\displaystyle\sum_{k=1}^n k^2$ equals $n/6+n^2/2+n^3/3$, but... why? And I also know that $\displaystyle\sum_{k=1}^n k^3$ equals $n^2/4+n^3/2+n^4/4$, but... is there a pattern so I can easily ...
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### Proof of correctness of Putzers algorithm

I have a question regarding the proof (seen below) of Putzers algorithm for matrix exponentiation. It's written by our danish lecturer at the university, so I translated the important parts into ...
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### Double summation including power and factorial [duplicate]

I am finding some trouble in computing the following sum: $$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac {y^m}{m!}$$ Could you please provide a result? Thanks in advance
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### Could $\sum e^{a_i}$ be simplified? Does it have an identity?

$\sum_{i=1}^n e^{a_i}$ (where $a_i \in \mathbb R$) is expensive for large $n$ (a sum and $n$ exponential operations). I was wondering if there is any way for simplifying this?
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### Bernoulli formula

The sum: $$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...+ n^m$$ Can be calculated by this formula, called the "Bernoulli formula" in wikipedia S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k ...
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### Summation Simplification

I am attempting to solve a problem that I posed myself, but I can't figure out how to simplify the solution from the "messy" state in which it currently exists. My mathematical background does not yet ...
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### Operators - sums, products, exponents, etc.

$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$. $(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$. Is there an operator, such that if ...
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### Computing $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$

I'm trying to find $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$ mod $m$. $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\varphi^3 = 2 + \sqrt{5}$. But honestly I'm not even sure where to start. ...
Sorry in advance, as I suspect I lack both the proper terms and the proper notation for the problem I have, but I'll try to be clear. If I have a set $S = \{1,2,3\}$, I figured out that the summation ...
### Summation over exponent $\sum_{i=0}^k 4^i= \frac{4^{k+1}-1}3$
Why does $\sum_{i=0}^k 4^i= \frac{4^{k+1}-1}3$, where does that 3 comes from? Ok, from your answers I looked it up on wikipedia Geometric Progression, but to derive the formula it says to multiply by ...