Questions tagged [summation]
Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
17,317
questions
0
votes
0
answers
18
views
$ p_k = \prod_{1 \leq j \neq k \leq n} |x_{k} - x_{j}|$ \ Prove that, $\sum_{k=1}^{n} \frac{1}{p_{k}} \geq 2^{(n - 2)}$
Let $n$ distinct points $i \in \mathbb{N}$ be on the interval $[1, -1]$. Define $p_{k}$ as,
$
p_k = \prod_{1 \leq j \neq k \leq n} |x_{k} - x_{j}|$
Prove that,
$
\sum_{k=1}^{n} \frac{1}{p_{k}} \geq 2^{...
- 386
1
vote
1
answer
53
views
Is there a name for this constant and its value where $\alpha = \sum_{p}\frac{\log \left({p}\right)}{p \left({p-1}\right)}$
The constant $$\alpha = \sum_{p} \frac{\log \left({p}\right)}{p \left({p-1}\right)}$$ comes from the calculation $$\sum_{p=2}^{x} \frac{\log \left({p}\right)}{p-1} = \sum_{p=2}^{x} \frac{\log \left({p}...
- 907
1
vote
1
answer
30
views
Question regarding portion of solution manual's argument to Chapter 19 Problem 36c of Spivak's Calculus
In Chapter 19 of Spivak's Calculus, Problem 36c)'s author-provided solution makes the following claim:
Supposing that $f$ is integrable on $[a,b]$, then, for any partition $\{t_0=a,t_1,\cdots,t_{n-1},...
- 4,802
0
votes
0
answers
24
views
Rewrite a summation in terms of three vectors
Suppose I have a summation $\sum_{i=1}^l a_i b_i c_i$, can I rewrite this in terms of three vectors $A=(a_1,\dots,a_l)^T, B=(b_1,\dots,b_l)^T, C=(c_1,\dots,c_l)^T$?
- 5
0
votes
0
answers
19
views
Can the odd values of this be vanished?
I'm looking for your help here:
For a central moment $M_n=\int_{-\pi /2}^{\pi /2}{\left( \theta -\theta _0 \right) ^ng\left( \theta ;\theta _0 \right) d\theta}$ about $\theta _0$, where $g\left( \...
0
votes
2
answers
40
views
$ \sum_{j=1}^{n} \sqrt[n]{\sum_{i=1}^{m} (x_{ji})^{n}} \geq \sum_{i=1}^{m} \left( \prod_{j=1}^{n} ( x_{ji}) \right)$
$$\forall x_{ij} \in \mathbb{R}^{+} , | , i \in {1, 2, 3, \ldots, m}, , j \in {1, 2, 3, \ldots, n} \Rightarrow \sum_{j=1}^{n} \sqrt[n]{\sum_{i=1}^{m} (x_{ji})^{n}} \geq
\sum_{i=1}^{m} \left( \prod_{j=...
- 386
5
votes
3
answers
125
views
Computation of $\displaystyle{\sum_{n=1}^{\infty}\frac{\sin nx \cdot \sin ny}{n^2}}$
First I used the identity
$$\sin nx \cdot \sin ny=\cos(n(x-y))-\cos(n(x+y))$$
and the sum turned into the following
$$\sum_{n=1}^{\infty}\frac{\sin nx \cdot \sin ny}{n^2}=\sum_{n=1}^{\infty}\frac{\cos(...
0
votes
0
answers
33
views
What is the name for a matrix which is generated by a recursive sum whose form equals a recursive product when replacing the sums with products?
In this answer to the question "Do these series converge to logarithms?" it is shown by George Lowther that each Dirichlet series involving the pattern of divisors converge to $\log(n)$ in ...
- 7,284
0
votes
0
answers
28
views
Can someone help me prove this statement by induction or some other proof method? [duplicate]
Show that $3n < 2^n$ for all integers $n \ge 4$. I have tried for base $n=5$: then obviously $p(5)$ is true as
$$
3\cdot 5 < 2^5 .
$$
Now since $p(x)$ is right I have tried to prove $p(x+1)$ by ...
3
votes
1
answer
52
views
Proof explanation of $\sum_{i=1}^{n} \lfloor\log\left(\frac{n}{i}\right)\rfloor$
$(1)$ The first relation.
$$
\left\lfloor \log_2\frac{n}{i} \right\rfloor=j
\Longleftrightarrow 2^j \leq \frac{n}{i} < 2^{j+1}
\Longleftrightarrow \frac{n}{2^{j+1}} < i \leq \frac{n}{2^j}
\...
- 49
-1
votes
1
answer
33
views
Does $f(i)+f(n-i)$ increase as $i$ decreases from $n/2$ to 1 for any superlinear function f?
I have a function f which is superlinear (for example $n\log n$ or $n^k$ for some constant $k$). Is true that for any such function we can prove that $f(i)+f(n-i)$ increases as $i$ decreases from $n/2$...
- 101
-1
votes
0
answers
54
views
Proving $3^{n-2}n(n-1)=\sum_{k=2}^n\binom{n}{k}k(k-1)\,2^{k-2} $ combinatorially
I am trying to find a combinatorial question that answers both sides of the equation but I am stuck.
$$
3^{n-2}n(n-1)=\sum_{k=2}^n\binom{n}{k}k(k-1)\,2^{k-2}
$$
original image
- 1
5
votes
0
answers
102
views
Compute $\sum^{2024}_{k=1} \frac{2023-2022 \cos \left(\frac{\pi(2k-1)}{2024} \right)}{2021-2020 \cos \left(\frac{\pi(2k-1)}{2024} \right)}$
Question: Compute $$\sum^{2024}_{k=1} \frac{2023-2022 \cos \left(\frac{\pi(2k-1)}{2024} \right)}{2021-2020 \cos \left(\frac{\pi(2k-1)}{2024} \right)}$$
I began by rearranging the sum as follows:
$$\...
- 483
-1
votes
0
answers
50
views
I am stuck here what next should I do please help me
$${prove}\:{that}\:\:\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\left(\mathrm{2}{r}\:+\:\mathrm{1}\right)\left(\:\overset{{n}} {\:}{C}_{{r}} \:\right)^{\mathrm{2}\:\:} =\:\left({n}+\mathrm{1}\...
- 9
1
vote
0
answers
43
views
Integration including the floor function
I'm aware of the way to quantify the following integral in the following way, however I'm trying to find another way to express the given integral. Especially when the function $f(x)$ is not ...
- 73
0
votes
0
answers
18
views
How can I represent a decreasing summation which the highest value is the starting value?
I am trying to find the recurrence of
$ T(n) = 2T(n-1) + n^2$ The answer it's $O(2^n n^2)$ but now I'm trying to find the answer using recursion and all good, but trying to find $T(n-k)$ is killing ...
-1
votes
0
answers
27
views
How to simplify this expression? ∑𝑛 𝑘=0 (𝑛 𝑘)/(k+1)
Ran into this task and got confused, how should I simplify it?
P.s. I'm sorry I'm still new here and don't really know how to add math formulas
1
vote
1
answer
68
views
Compute Sum : $\sum_{n＝1}^{\infty}\frac {1}{n}\Big(\frac {\ln(n)}{n－\ln(n)}\Big)$
Compute Sum : $\sum_{n＝1}^{\infty}\frac {1}{n}\Big(\frac {\ln(n)}{n－\ln(n)}\Big)$
This problem is from the book :Book proplem analysis I
proplem 3.1.5 (b)
The solution from the book:
My question:...
- 832
1
vote
0
answers
91
views
solving question on recurrence relation and sumation
Let $\{x_{n}\}$ be a sequence of non-negative real numbers such that $x_{n + 1}^2 =6x_{n} +7$ for all $n ≥ 2$
Which one of the following is true?
$\quad(A).\space$ If $x_{2} > x_{1} > 7$ then $\...
0
votes
1
answer
115
views
Is there an easy way to calculate this infinite summation
Is there an easy way to calculate this summation of integral:
$$\sum_{n=0}^\infty \int_{r=0}^1 \frac {(r-\frac{1}{2})\cos(c\cdot\ln(r+n))} {(r+n)^{1-b}} dr $$
The most obvious approach is to calculate ...
0
votes
1
answer
59
views
Simplify Summation $S_n=\sum_{k=1}^{8n} (-1)^\frac{(k)(k+1)(k+2)}{6}(k)^2+\sum_{k=1}^{8n}(-1)^\frac{(k+2)(k+3)}{2} (k)^2-4\sum_{k=1}^{8n}(8k-2)^2$
If $$S_n = \sum_{k=1}^{8n} (-1)^\frac{(k)(k+1)(k+2)}{6} (k)^2 + \sum_{k=1}^{8n} (-1)^\frac{(k+2)(k+3)}{2} (k)^2 -4\sum_{k=1}^{8n} (8k-2)^2$$
then the value of $-S_{40}$ is equal to?
Simplifying all 3 ...
- 41
0
votes
1
answer
27
views
Directional cosine of normal unit vector
In below directional cosine of binormal unit vector instead of $${l} = {y'z''-z'y''}/ \sqrt{(y'z''-z'y'')^2+(y'x''-x'z'')^2+(y'z''-z'y'')^2}$$ and similarly m and n DC.
Following term are used using ...
0
votes
0
answers
22
views
Empty sum of undefined function
An undergraduate real analysis homework problem I am working on raised the following question:
Does function $f$ need to be defined for $i=0,1$ for empty sum $\sum_{i=1}^0{f(i)}$ to be equal to zero? ...
- 160
1
vote
1
answer
48
views
Trying to understand this skipping sum
$$ f(x) = \begin{cases} 0 & x \not\equiv 4 \pmod 5 \\ 1 & x \equiv 4 \pmod 5 \end{cases} $$
and $$ f(x) = \frac{1}{5} \sum_{k=0}^4 \cos\left(\frac{2 \pi}{5} k (x-4) \right) $$
Can someone help ...
- 420
1
vote
1
answer
61
views
Asymptotic equivalent of $\sum_{k=1}^n a^k k^{-1/2}$
I encountered recently the following partial sum $\sum_{k=1}^n a^k k^{-1/2}$ with $a$ a constant approximately equal to $2.955$.
I was wondering if there were any clever way to find an asymptotic to ...
- 680
0
votes
1
answer
48
views
Is $\lim_{m\to\infty}\sum_{k≥t}\frac{1}{\binom kt^m}=1$?
Context:
Using Wolfram calculator, I've observed that :
$$\sum_{k≥2}\frac{1}{\binom k2^{100}}≈1$$
$$\sum_{k≥5}\frac{1}{\binom k5^{100}}≈1$$
$$\sum_{k≥4}\frac{1}{\binom k4^{50}}≈1$$
Question:
I want to ...
- 2,009
3
votes
0
answers
62
views
Is $1105$ the only Poulet-number of the form $2^a+3^b$?
Conjecture : $1105$ is the only Poulet-number that can be written in the form $2^a+3^b$ with positive integers $a,b$
A Poulet-number is a composite number $N$ satisfying the congruence $2^{N-1}\equiv ...
- 82.2k
0
votes
1
answer
53
views
To prove an inequality related to the proximal operator (or non-expensive operator)
I am seeking assistance in proving an inequality that I believe holds for a specific mathematical concept involving the proximal operator based on a proper convex function.
The inequality is as ...
0
votes
0
answers
42
views
Is this sum right represented for the case $x=0$?
if $g(x)=\sum_{n=0}^{\infty} \binom{k}{n} x^n $ then $g(0)=1$ ?
I am confused, isn't it supposed to yield $0^0$ as a factor in the first term (which would be an indefinition at least, for this context)...
- 15
0
votes
0
answers
46
views
Summation with inner products: properties and rearrangement
OPTION 1.
I have this expression,
$$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l<j}^{L}a_l^2 \frac{a_j}{a_l}$$
and I would like to take out $$\sum \limits_{l=1}^{L}a_l^2$$, as to obtain something ...
- 131
1
vote
0
answers
19
views
Partition of n into k parts with at most m
I ran into a problem in evaluating a sum over kronecker delta. I want to evaluate
$$\sum_{\ell_1,...,\ell_{2m}=1}^s\delta_{\ell_1+\ell_3+...+\ell_{2m-1},\ell_2+\ell_4+...+\ell_{2m}}$$
My approach was ...
- 29
1
vote
0
answers
35
views
Polylogarithm further generalized
Here I proposed a generalized formula for the polylogarithm. However, because of a slight mistake towards the end, visible prior to the edit, I was unaware that it yields just a result of an integral ...
- 979
1
vote
0
answers
61
views
Proof of Cesàro summation
This is a proof I came up with while working on the textbook Understanding Analysis:
Supposing $x_{n} \rightarrow x$, we have that
$$s_n = \frac{1}{n} \sum_{k=1}^{n} x_k \rightarrow x$$
Let $\epsilon \...
- 349
0
votes
0
answers
28
views
Distributivity property of summations when index is equal among summations
I Know that this equivalence is true, considering the distributivity property of summation:
$$\sum \limits_{i=1}^{N}x_i g_{i,l}\sum \limits_{j=1}^{N}x_j g_{j,k}=\sum \limits_{i=1}^{N}\sum \limits_{j=1}...
- 131
2
votes
2
answers
87
views
How to prove that the above summation is equal to $(\pi/101)\tan(\pi/202)$
$$\sum_{n=0}^\infty\left(\frac1{50+101n}-\frac1{51+101n}\right)$$
How to prove that the value of the above summation is equal to $(\pi/101)\tan(\pi/202)?$ I am trying this question by putting n=0,1,2,...
1
vote
1
answer
56
views
Question on integer values of Riemann's prime-power counting function
Is it known whether Riemann's prime-power counting function
$$\Pi(x)=\sum\limits_{p^n\le x} \frac{1}{n}=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\log(n)}=\sum\limits_{n=1}^{\log_2(x)} \frac{1}{n}\, \pi\...
- 6,693
-4
votes
0
answers
59
views
Summation of infinite diverging sequence [closed]
Please help me solve this
\begin{equation}
\sum_{n=1}^\infty \frac{1}{4n^4+1} = ?
\end{equation}
- 9
0
votes
1
answer
65
views
Closed form solution for partial summation of $\sum_{x=1}^{k} \frac{2^{\frac{1}{x}}}{{x^2}}$
Recently I've been working on solving summations and I found this one to be quite tricky.
$\sum_{x=1}^{k} \frac{2^{\frac{1}{x}}}{{x^2}}$
The integral which this is based off of, can be solved with u ...
0
votes
0
answers
17
views
The Proof by Induction of the Multinomial Theorem
I looked at the proof by induction of the multinomial theorem on Wikipedia and do not understand how to get the last step. Specifically, I do not know why this equality is true:
$$\sum_{k_1 + k_2 + \...
2
votes
0
answers
62
views
Evaluating $\sum_{k=0}^{\infty} \frac {2^k}{5^{2^k}+1}$ [duplicate]
So my teacher shared this problem with us and said everyone needs to try this, he teaches us Olympiad Math so I am assuming this wouldn't require analysis or calculus. This is the question, I have ...
- 51
0
votes
0
answers
94
views
have to prove that these two binomial series are equal
Question:
We have to prove that the following two binomial series are equal
$$\sum_{r=1}^k2^r\binom nr\binom{k-1}r =\sum_{r=1}^k\binom nk\binom{n+k-r-1}{n-1}.$$
I have tried expanding the binomial ...
4
votes
0
answers
88
views
Confinement result for unit complex numbers
Inspired by this problem, and some computer simulations, I almost convinced myself of the following result. However, I am coming short on a proof.
Result: Let $n\geq 3$, and $2n+1$ complex numbers $z_{...
- 2,203
0
votes
1
answer
58
views
How to find the summation of the above series up to infinity?
How to find the summation of the above series up to infinity? I have tried to find the above summation by putting n=0,1,2,3,...and so on. Finally I got the above series to be
(1-(1/2)+(1/23)-(1/24)+(1/...
1
vote
1
answer
75
views
Summation of Alternating Products in Pascal's Triangle: prove $\sum_{k=1}^{n} (-1)^{k+1} x_{k} y_{n-k+1} = 1$ [duplicate]
Dappy is playing around with Pascal's Triangle. While looking for patterns, he finds one equality that he likes a lot:
Pick any spot on Pascal's Triangle that isn't part of the right edge. Call the ...
- 386
1
vote
1
answer
29
views
Approximate summation formula of time to count numbers from 1 to N
When calculating how much time it takes to count from $1$ to $n$, it is normally used the approximation that it takes about $1s$ to say a number out loud, so it would take $n$ seconds, but there's a ...
- 481
-2
votes
0
answers
69
views
How could I correct the increase of 10 in this formula? [closed]
30 * (x-3)+30
The results are:
#x = 3 el resultado es 30
#x = 4 el resultado es 60
#x = 5 el resultado es 90
#x = 6 el resultado es 120
#x = 7 el resultado es 150
#x = 8 el resultado es 180
#x = 9 el ...
1
vote
1
answer
62
views
Evaluate $ \sum_{k=0}^{\infty} \frac{2^{k+7}}{5^{2^{k}} + 1} $
Evaluate $ \sum_{k=0}^{\infty} \frac{2^{k+7}}{5^{2^{k}} + 1} $
For context, I encountered this question in a recent multiple choice examination.
Here is my solution for this question:-
Taking $2^{7}$ ...
- 56
0
votes
0
answers
27
views
Merging a double summation into a single one
I'm looking for merging this summations into a single one:
$\displaystyle \sum_{i=0}^k{\sum_{j=0}^i{b^j}}$ where $b$ is an integer.
I know it is equal to:
$\displaystyle \sum_{i=0}^k{b^i(k-i+1)}$
In ...
- 45
0
votes
1
answer
42
views
If I roll a 6 sided die infinite times, each time calculating the sum of all rolls. What is the probability the sum will be 2023 after a certain roll. [closed]
So after each time you roll the dice you find the sum of the current roll and the previous rolls. As this number get larger by 1 to 6 each time will it will pass 2023 at some point. What is the ...
1
vote
2
answers
89
views
$\lim_{{n \to \infty}} \sum_{{k=1}}^{n} \arctan\left(\frac{1}{k}\right) - \ln n$
$\arctan(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{2m+1}x^{2m+1}$
\begin{align*}
\arctan\left(\frac{1}{k}\right) &= \sum_{m=0}^{\infty} \frac{(-1)^m}{2m+1}\left(\frac{1}{k^{2m+1}}\right)
&= \frac{...
- 386