All Questions
96
questions
1
vote
2
answers
239
views
Formula for the general Cavalieri Sum: $S_n(p)=\sum\limits_{k=1}^{n} k^p\,\,\,n, p\in\mathbb N$ [duplicate]
What kind of formula is there that can be used for calculating the sum of power of $x$ of numbers from $1$ to $a$?
I know that the sum of numbers from $1$ to $a$ is $\ (n^2 + n)/ 2 \ $ and that the ...
- 111
0
votes
1
answer
2k
views
Range of values of x for the sum to be valid.
Given the series:
$$\frac{1}{x+1}+\frac{1}{(x+1)^2}+\frac{1}{(x+1)^3}+...$$
Find the sum to infinity for the series and state the range(s) of values of $x$ for the sum to be valid.
I solved the ...
- 7,798
1
vote
1
answer
56
views
Given variable $m$, how do I find zeros of a polynomial in terms of $m$?
This is a summation question about a finite series with sum $m$. I'm trying to write a computer program that takes in a given integer $m$ (which represents the sum of a series) and outputs the number ...
- 11
1
vote
1
answer
83
views
How prove this identity $\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\min{\{i,j\}}(a_{i}-a_{i+1})(b_{j}-b_{j+1})=\sum\limits_{i=1}^{n}a_{i}b_{i}$?
Show that this identity;
$$\sum_{i=1}^{n}\sum_{j=1}^{n}\min{\{i,j\}}(a_{i}-a_{i+1})(b_{j}-b_{j+1})=\sum_{i=1}^{n}a_{i}b_{i}$$
where $a_{i},b_{i}\in R,(i=1,2,\cdots,n),a_{n+1}=b_{n+1}=0$
It seem can ...
- 93.6k
3
votes
2
answers
214
views
Prove with sequence
$(a_n)^{\infty}_{n=0}$ is a sequence that define by
$$a_n=\begin{cases}\frac{n}{2} & \mathrm{if} & n=2k\\\frac{n-1}{2} & \mathrm{if}& n=2k+1\end{cases}$$
suppose $S(n)=a_0+a_1+a_2+\...
- 25.2k
1
vote
1
answer
110
views
Show $ \sum\limits_{k=1}^n \frac{a}{n}\cos(\frac{ka}{n})=\frac{\frac{a}{2n}}{\sin(\frac{a}{2n})}\sin(a+\frac{a}{2n})-\frac{a}{2n} $
I already got the worst part figured out through a proof using the complex definition of sine, I got
$$ \sum\limits_{k=1}^n \frac{a}{n}\cos(\frac{ka}{n})= \frac{a}{n}(\frac{\sin(n+\frac{1}{2})\frac{a}{...
- 3,012
2
votes
6
answers
2k
views
Finite sum $\sum_{j=0}^{n-1} j^2$
How can I calculate this finite sum? Can someone help me?
$$\sum_{j=0}^{n-1} j^2$$
user335080
0
votes
1
answer
51
views
We have an ISBN: $\left(\sum_{k=1}^{9}(11-k) \cdot x_{k}+p\right) \text{ mod } 11=0$
$x_{1},...,x_{9} \in \left\{0,1,...,9\right\}$ are the first nine
places of an ISBN and $p$ is the check digit, given by these digits
which fulfills the condition:
$\left(\sum_{k=1}^{9}(11-k) ...
- 375
0
votes
0
answers
66
views
Limit in sum and fraction
I'm staring at
$$ f(n) = \sum_{x=1}^n a(n)^{n-x}\\
a(n) = \frac{(1 - \frac{f}{n})y}{y(1 - \frac{f}{n} - \frac{\delta}{n}) + \frac{\delta}{n}}$$
where $f$, $y$, $\delta$, all $ \in (0, 1)$, and $n$ ...
- 1,089
1
vote
1
answer
63
views
Reducing Poisson Sum expression
I am trying to reduce and obtain a simple expression from following sum
$$
\sum_{i = 1}^{\infty}{1 \over i}\,
{\,\mathrm{e}^{-\lambda}\,\lambda^{i} \over i!}\,
{1 \over 1 - \,\mathrm{e}^{-\lambda}\,}
...
- 97
1
vote
0
answers
42
views
Evaluating a limit involving summation [duplicate]
Evaluate :
$$ \lim_{n\to\infty}\left(\dfrac{1}{e^{n}}\displaystyle \sum_{r=0}^{n} \dfrac{n^r}{r!}\right) $$
Numerical calculation suggests that the limit should be $\dfrac{1}{2}$. I tried using ...
- 5,719
-4
votes
2
answers
110
views
proving $\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$ [closed]
how can one prove that:
$\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$
For all natural $n$,
without using induction?
thank you.
- 77
-1
votes
2
answers
51
views
are these summations equal
Give Function A:
$$ \frac{1}{2} \gamma X^2 - \frac{1}{2}\gamma \sum_{i=1}^N n_k^2
$$
and Function B:
$$ \epsilon \sum_{i=1}^N |n_i| + \frac{\eta}{\tau} \sum_{i=1}^N {n_i}^2$$
Can you show that ...
- 123
4
votes
3
answers
607
views
Upper bound for $\sum_{n=1}^xn^{k-1}$
From some Calculus and guess-work, I found that
$$k\sum_{n=1}^xn^{k-1}<(x+\frac12)^k\tag1$$
In fact, I found that it was very, very, close.
And, from even more guesswork,
$$\lfloor(x+\frac12)^k\...
3
votes
0
answers
171
views
How to prove the identity $\sum_{n=1}^{\infty} \dfrac{{H_{n}}^2}{n^2} = \dfrac{17}{360} {\pi}^4$? [duplicate]
Prove That
$$\sum_{n=1}^{\infty} \dfrac{{H_{n}}^2}{n^2} = \dfrac{17}{360} {\pi}^4$$
I encountered this identity while reading the article about Harmonic Number on Wikipedia. I thought of using the ...
user313478