18
votes
4answers
291 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
1
vote
0answers
61 views

Rising/Falling Powers, Summation

1) Show that $$(-n)^{\bar p} = (-1)^n n^\underline{p}$$ (original screenshot) 2) Evaluate the sum $$\sum_{a\le n\lt b}n^{\bar p}$$ (original screenshot) Thoughts regarding question 1: I've ...
0
votes
1answer
17 views

Exponential equivalent for geometric space

I'm just starting a foray into geometric algebra and calculus so that I can develop a geometric version of the standard arithmetic neural net. Specifically when calculating the error function for a ...
1
vote
1answer
45 views

Is there a formula for a sequence like $k^{t}-k^{t-1}+k^{t-2}-…+k^{2}-k^{1}+k^{0}$

I am trying to solve a programming problem and my intended solution involves a calculation like this one: $k^{t}-k^{t-1}+k^{t-2}-...+k^{2}-k^{1}+k^{0}$ The problem is that $t$ can be as large as ...
1
vote
3answers
43 views

What is the summation of the following expression?

What's the summation of the following expression; $$\sum_{k=1}^{n+3}\left(\frac{1}{2}\right)^{k}\left(\frac{1}{4}\right)^{n-k}$$ The solution is said to $$2\left(\frac{1}{4} ...
3
votes
2answers
153 views

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 ...
4
votes
2answers
111 views

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 ...
2
votes
1answer
78 views

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 ...
1
vote
0answers
105 views

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
0
votes
1answer
64 views

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?
3
votes
0answers
113 views

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 ...
4
votes
4answers
334 views

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 ...
4
votes
1answer
146 views

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 ...
0
votes
2answers
65 views

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. ...
3
votes
1answer
141 views

Summing 2 to the power of the subset sums of a power set

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 ...
3
votes
5answers
163 views

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 ...