All Questions
Tagged with summation algebra-precalculus
90
questions with no upvoted or accepted answers
9
votes
0
answers
299
views
Set $S$ of integers $\ge0$ such that $\{1,2,\cdots,n\}$ partitions into $A, B$ with $\sum_\limits{a\in A}a^k=\sum_\limits{b\in B}b^k$ for all $k\in S$
Let $s_n$ be the largest size of a set $S$ of integers $\ge0$ st there exist two subsets $A,B\subseteq \{1,2,...,n\}$ that satisfy the conditions
(1) $A\cap B=\emptyset$,
(2) $A\cup B=\{1,2,...,n\}$,
(...
- 184
6
votes
1
answer
143
views
Equality of Floors of some Partial Sums
Let $S_n=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}$ denote the $(n+1)^{st}$ partial sum in the series expansion for $e=\sum_{k\ge 0}\frac{1}{k!}$. I want to prove that $\lfloor n\cdot(...
- 61
4
votes
0
answers
306
views
Closed form for Sum of Tangents with Angles in Arithmetic Progression
The formulae that can be used to evaluate series of sines and cosines of angles in arithmetic progressions are well known:
$$\sum_{k=0}^{n-1}\cos (a+k d) =\frac{\sin( \frac{nd}{2})}{\sin ( \frac{d}{2} ...
- 6,560
4
votes
2
answers
78
views
Is $\left(\sum_{n=1}^N\frac{a_n}{N}\right)^N\left(\sum_{n=N+1}^{2N}\frac{a_n}{N}\right)^N≠\left(\sum_{n=1}^{2N}\frac{a_n}{2N}\right)^{2N}$?
Let
$$G_N= \prod_{n=1}^Na_n$$
and
$$A_N=\left(\frac{\sum_{n=1}^Na_n}{N}\right)$$
So
$$G_{2N}= \prod_{n=1}^{2N}a_n \\
=\left(\prod_{n=1}^{N}a_n\right)\left(\prod_{n=N+1}^{2N}a_n\right) \\
≤_{IH}\left(\...
- 1,359
4
votes
0
answers
54
views
Is there any error in my solution : If $\sum^n_{r=1} r^4=I(n), $ then $\sum^n_{r=1}(2r-1)^4$ is equal to ..
Problem :
If $\sum^n_{r=1} r^4=I(n), $ then $\sum^n_{r=1}(2r-1)^4$ is equal to
(a) $I(2n)-16I(n)$
(b) $I(3n)-2I(n)$
(c) $I(2n)-I(n)$
(d) $I(2n)+I(n)$
Please suggest if there is some error ...
- 10k
3
votes
2
answers
117
views
$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$
$$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$$
I solved it by letting the sum be $S$, then adding the sum to itself but taking the terms from last to first and then ...
3
votes
0
answers
100
views
Simplifying a subindex equation
Consider the following equation
$$
\frac{e^{\sum_i \alpha_i x_i}}{\sum_i x_i}=\sum_k\frac{e^{\sum_i \alpha_i y_{i, k}}}{\sum_i y_{i, k}}
$$
Is it possible to write $x_i$ as a function of the terms $y_{...
- 3,435
3
votes
1
answer
46
views
Rewriting a system of two equations found in The Chemical Basis of Morphogenesis (A. Turing, 1952)
In reading The Chemical Basis of Morphogenesis
by A. Turing, I am unable to follow a small section of his working. On page $47$, Turing states that
\begin{align*} x_r&=\sum_{s=0}^{N-1} \exp\left(\...
- 617
3
votes
0
answers
30
views
for what value of $y$ does $\sum_{k= 0}^n a_0 x^k = \sum_{k=0}^n y^k$
for what value of $y$ does $$\sum_{k= 0}^n a_0 x^k = \sum_{k=0}^n y^k$$
This was just an idea I was playing around with. I tried solving
$$\frac{a_0(x^{n+1}-1)}{x-1} = \frac{y^{n+1}-1}{y-1} $$
This ...
- 4,844
3
votes
0
answers
287
views
$f(8) \geq 1$ and $f(n)\geq 2f(\lceil \frac n2-n^{2/3} \rceil)$. Can we deduce $\exists C>0: f(cn) \geq n$?
Let $f : \Bbb N \to \Bbb N$ be a nondecreasing function that satisfies $f(8) \geq 1$ and $f(n)\geq 2f(\lceil \frac n2-n^{2/3} \rceil)$. Can we deduce that there exists some positive constant $c$ such ...
- 929
3
votes
0
answers
68
views
How many tests to validate an identity?
Discrete formulas such as the Faulhaber summations can be verified by evaluating them for a finite number of values.
For example $$\sum_{k=1}^nk=\frac{n(n+1)}2$$ is validated by evaluating for $n=0,1,...
user65203
3
votes
0
answers
283
views
Pull constant out of a summation of fractions
General problem
$$
\sum_{i=1}^n \frac{a_i + x}{b_i + x} = 0
$$
Is it possible for solve for $x$?
Some context
I've hit a road block in my derivation... At this point, I need to pull the model ...
- 31
3
votes
2
answers
176
views
Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$
Evaluate
$$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$
where $i=\sqrt{-1}$
For this question, I did the following,
Let
$$
\begin{align*}
S &= \sum_{n=1}^{\infty} \...
user208998
2
votes
0
answers
60
views
How to simplify $e^{-\sum_{i=-k}^k(k-|i|)x_i}$?
Consider the expression given by
$$
\large e^{-\Large\sum_{i=-k}^k(k-|i|)x_i}
$$
Is there a way of simplifying this expression?
For example, provided $\{x_i\}$ is bounded and "smooth" enough ...
- 3,435
2
votes
0
answers
39
views
Simplifying $\sum_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$ in two ways gives different results
I want to calculate the result of
$$\sum\limits_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$$ I used two below approaches. Both approaches are based on $\log A + \log B = \log (A \times B)$ and $\sum\...
- 485
2
votes
0
answers
78
views
value of $\frac{\sum_{k=0}^r{n\choose k}{n-2k\choose r-k}}{\sum_{k=r}^n {n\choose k}{2k\choose 2r} {(\frac{3}{4})}^{(n-k)}({\frac{1}{2}})^{2k-2r}}$ .
The question requires us to find the value of $\frac{\sum_{k=0}^r{n\choose k}{n-2k\choose r-k}}{\sum_{k=r}^n
{n\choose k}{2k\choose 2r} {\left(\frac{3}{4}\right)}^{(n-k)}\left({\frac{1}{2}}\right)^{...
- 97
2
votes
0
answers
60
views
Finding a formula for a sum that involves binomial coefficients
Is there a formula for this sum:
$$ \sum_{j=0}^k {n \choose j} {n \choose k-j} (-2)^j \left(-\frac13 \right)^{k-j} ?$$
It reminds me to Vandermonde's identity; but as you can see there is a slight ...
- 897
2
votes
0
answers
143
views
I've come up with two ways to evaluate $\sum_{1 \le j<k\le n} \frac{k}{k-j}$ but only one of them works
I've come up with two ways to solve this double sum but only one of them works:
$$ \sum_{1 \le j<k\le n} \frac{k}{k-j}$$
My first approach is to change $k-j$ into a single $k$. So we have the ...
2
votes
2
answers
82
views
Using trigonometric power formulas to derive an identity for $\cos^3(x)$
I am practicing with manipulating sigma notation and binomial coefficients right now. I am using the formula given here to derive the identity for $\cos^3(x)$
The identity for $\cos^3(x)$ is
$$\cos^3(...
- 3,187
2
votes
0
answers
137
views
Can this summation be done without calculator?
Is it possible to perform the summation ,
$$\sum_{i=1}^{\infty} \frac{1}{i^i}$$
without the use of calculator?
It does converge to a finite value = 1.29129...
Wolfram Alpha link to this
Describe the ...
- 1,663
2
votes
0
answers
115
views
Double Summation Multiplication
There is some simplification, similar to Lagrange's identity, for the multiplication of double summation ?
Double Summation:
$\left( \sum\limits_{\substack{m=1}}^N \sum\limits_{\substack{n=1}}^N a_{...
- 63
2
votes
0
answers
38
views
Upper bound on $\frac{v_{1}}{1-c_{1}x}+\cdots+\frac{v_{n}}{1-c_{n}x}$
Let $v_{1},\ldots,v_{n}$ and $c_{1},\ldots,c_{n}$ be real numbers such that $v_{i}=2\alpha_{i}^{2}$ and $c_{i}=2\alpha_{i}$ for some $\alpha\ge 0$. My question is the following: Can I get, for $x\ge 0$...
- 638
2
votes
0
answers
256
views
Foil in a Summation
I have the following summation, where I find the following result
$ \sum_{i}^n {(a+b_i)(c+d_i)} $
$ \sum_{i}^n {(ac+ad_i + bc_i + b_id_i)} $
However others have told me that I am missing a "N" ...
- 85
2
votes
2
answers
70
views
Question on changing the index of summation
$$b(a+b)^m = \sum_{j=0}^m \binom{m}{j}a^{m-j}b^{j+1}= \sum_{k=1}^m \binom{m}{k-1}a^{m+1-k}b^{k}+b^{m+1}$$
I believe $j = k-1$ though the book does say that.
This is related to proving the binomial ...
- 21
2
votes
0
answers
150
views
Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$
For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate
$n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$
So the product goes up to $k$ and I ...
- 69
2
votes
0
answers
105
views
How does one change the top number in a summation?
Sorry I do not know the correct term (I am guessing "upper limit"). Here is what I mean.
$$\sum\limits_{i=1}^{\color{red}{17}}\frac{2i}{i+3}$$
The $17$ is what I am talking about as "the top number". ...
- 5,659
2
votes
0
answers
48
views
Are there efficient ways of computing sums that involve trigonometric functions and q-logarithms (Tsallis q-logarithms)?
I am interested in computing the following sum:
\begin{equation}
\sum\limits_{l=1}^k l^{\beta_1} \cos\left(\omega \log_q(\frac{l}{t_c})\right)
\end{equation}
Here $0 < \omega$, $0 < k < ...
- 11.5k
2
votes
0
answers
730
views
Analytical solution for a variable inside of a summation
I am trying to figure out how to solve the following expression for $x$ and I'm surprised that I don't know what to do.
$$\frac{2n}{x} = \sum_{i=1}^{n} \frac{1}{x-y_{i}}$$
We have that $n$ and $x$ ...
- 5,102
1
vote
0
answers
137
views
Simple algebra in rearring terms
I have a very simple mathematical question, and it is just about algebra which seems very tedious. First, let me state my problem from the beginning:
Let $i$ be an index representing countries ($i = {...
- 105
1
vote
0
answers
103
views
Restructuring Jacobi-Anger Expansion
In Jacobi-Anger expansion, $$e^{\iota z \sin(\theta)}$$ can be written as:
$$e^{\iota z \sin(\theta)} = \sum_{n=-\infty}^{\infty} J_n(z)e^{\iota n \theta}$$
where $J_n(z)$ is the Bessel function of ...
- 31
1
vote
0
answers
92
views
Example of a specific polynomial
I need four non-trivial polynomials $P(x)$, $Q(y)$, $R(z)$ and $S(w)$ such that $$P(x)Q(y)R(z)S(w)=\sum_{i}a_i b_i c_i d_i x^i y^i z^i w^i+ \sum_{j\neq k\neq l\neq m}e_j f_k g_l h_m x^j y^k z^l w^m $$ ...
- 918
1
vote
0
answers
47
views
Multiple Sigma Notation and Expected Value
I've been struggling with expected value calculations that involve multiple or nested sigma notations. For instance, the solution to a problem I was working on is:
$X=\Sigma_{i=1}^{10}X_i\implies E[X]...
- 111
1
vote
0
answers
39
views
Is this a valid proof method (and how to complete it) of the exchange of order of indexed summations?
I want to prove that
$$
\sum_{1 \leqq i \leqq a} \sum_{1 \leqq j \leqq b} f(i, j) = \sum_{1 \leqq j \leqq b}\sum_{1 \leqq i \leqq a} f(i, j)
$$
Which is a very simple statement, but also a bit vexing ...
- 193
1
vote
0
answers
139
views
Solving a geometric-harmonic series
Find the value of $\displaystyle \frac21- \frac{2^3}{3^2}+ \frac{2^5}{5^2}- \frac{2^7}{7^2}+ \cdots$ till infinite terms.
found this problem while integrating $\arctan\left(x\right)/x$ from $0$ to $2$ ...
1
vote
0
answers
86
views
Limit of a sum and two ways yielding two answers
$$\lim_ {n\to \infty} \sum_{r=0}^n \frac{1}{(2r+1)(2r+3)(2r+2)}$$
Now, what I did here was to first break up the general term of the sum using partial fractions, yielding
$$ \frac{1}{(2r+1)(2r+2)(2r+3)...
- 33
1
vote
0
answers
40
views
Relating $\sum_{k=1}^N a_k^2 e^{\frac{2\pi i}{N}k}$ to $(\;\sum_{k=1}^N a_k e^{\frac{2\pi i}{N}k}\;)^2$
Consider the following expression
$$
\sum_{k=1}^N a_k^2 e^{\frac{2\pi i}{N}k}\tag{1}
$$
where $i$ is the imaginary number. How may I relate it to the following expression
$$
\left(\sum_{k=1}^N a_k e^{\...
- 3,435
1
vote
0
answers
48
views
seperating two variables in a function with summation
I'm building a data analysis program that perform on big chunks of data, the issue I'm having is the speed of some operations; to be exact I have a function that takes two variables in this form : $$f(...
- 113
1
vote
0
answers
46
views
can anyone simplify this equation? $\sum_{n=2}^{\infty} \sum_{j=1}^{n}\left[\frac{n !}{j !(n-j) !}(-A)^{n-j}(-B)^{j}\right]$
Can anyone help me simplify the following equation? Any ideas appreicate. Thanks a lot!
$\sum_{n=2}^{\infty} \sum_{j=1}^{n}\left[\frac{n !}{j !(n-j) !}(-A)^{n-j}(-B)^{j}\right]$
with two variables A ...
- 31
1
vote
1
answer
194
views
Trigonometric Identities Using De Moivre's Theorem
I am familiar with solving trigonometric identities using De Moivre's Theorem, where only $\sin(x)$ and $\cos(x)$ terms are involved. But could not use it to solve identities involving other ratios. ...
1
vote
1
answer
45
views
Sum and product problem
How can I find the result of:
$\sum\limits_{i=1}^{n}\prod\limits_{j=1}^{2} ij$
I know that $\prod\limits_{j=1}^{2} ij = 2i^2$, so I should simply do the summation as $\sum\limits_{i=1}^n2i^2$?
1
vote
0
answers
56
views
Prove that if $4k+1$ is a prime number than for every $n \in \mathbb{N}$ $\sum\limits_{j=1}^{2k}{j^{4n+2}}$ is a multiple of $4k+1$
Prove that if $4k+1$ is a prime number than for every $n \in \mathbb{N}$ $\sum\limits_{j=1}^{2k}{j^{4n+2}}$ is a multiple of $4k+1$
First I tried proving for n=1
$\sum\limits_{j=1}^{2k}{j^{6}}=(4k+1)...
- 191
1
vote
0
answers
28
views
How to express this function for arbitrary N?
I am working on solving an equation system, that involves a parameter $N=2,3,\cdots,\infty$. Because I found it rather hard to solve it with an arbitrary $N$, I solved it in smaller numbers. One can ...
- 11
1
vote
0
answers
75
views
When is $a(n)$ prime?
Question: When is $a(n)\in P$ compared to all possible values of $n$? where $P$ denotes the set of primes. What is the density of the primes in the sequence?
Consider the sum of the prime counting ...
- 126
1
vote
0
answers
57
views
Closed form of basic looking geometric-like sum
In my research I have come across the need to sum simple looking geometric series-like sums. Neither Maple, nor Mathematica, nor Wolfram Alpha and not even OEIS (On-line Encyclopedia of Integer ...
- 11
1
vote
0
answers
48
views
Transforming a long sum of products for efficient computation (based on spanning trees in a complete graph)
Let's say there is a set of $n$ real coefficients: $a_1,...,a_n$. My task is to calculate the value of a rather simple sum of k products: ...
- 239
1
vote
0
answers
46
views
Trying to find a function that converts a binary sequence to a number
Suppose I have the sequence $\langle x_1, ..., x_n \rangle \in \mathbb{N}^n$ that gets encoded to the binary string
$$
\begin{matrix}
\underbrace{11...1} & 0 & \underbrace{11...1} & 0 &...
- 978
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
116
views
Upper bound of a sum of series
How can I find a tight upper bound for the following expression:
$\sum\limits_{i=1}^{k} a_i \sum\limits_{j = 1}^{i} \frac{1}{b_j} = a_1 \frac{1}{b_1} + a_2 (\frac{1}{b_1} + \frac{1}{b_2}) + \dots + ...
- 41
1
vote
0
answers
35
views
Help with simplification of this expression.
So i have derived this expression and would like to simplify it (i.e find an expression purely in terms of J and v).
$$ J\bigg[1+\sum_{n=1}^{J-1}\prod_{k=1}^{n}\frac{k(1-v)(J-k)}{(k+1)(J-k-1 + vk)}\...
- 61
1
vote
0
answers
160
views
How to isolate and solve for k in a Sigma notation probability mass function equation?
"isolate and solve for k:"
$$P(X = k) = \sum_{k=0}^n {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$$
If the above equation is a function of P, how would the equation be stated as a ...
- 11