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) $A\cup B=\{1,2,...,n\}$,
(...
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(...
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:
$$\sum_{k=0}^{n-1}\cos (a+k d) =\frac{\sin( \frac{nd}{2})}{\sin ( \frac{d}{2} ...
4
votes
2
answers
78
views
Is $\left(\sum_{n=1}^N\frac{a_n}{N}\right)^N\left(\sum_{n=N+1}^{2N}\frac{a_n}{N}\right)^N≠\left(\sum_{n=1}^{2N}\frac{a_n}{2N}\right)^{2N}$?
Let
$$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(\...
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 ...
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
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(\...
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 ...
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 ...
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,...
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 ...
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
$$
\begin{align*}
S &= \sum_{n=1}^{\infty} \...
2
votes
0
answers
60
views
How to simplify $e^{-\sum_{i=-k}^k(k-|i|)x_i}$?
Consider the expression given by
$$
\large e^{-\Large\sum_{i=-k}^k(k-|i|)x_i}
$$
Is there a way of simplifying this expression?
For example, provided $\{x_i\}$ is bounded and "smooth" enough ...
2
votes
0
answers
39
views
Simplifying $\sum_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$ in two ways gives different results
I want to calculate the result of
$$\sum\limits_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$$ I used two below approaches. Both approaches are based on $\log A + \log B = \log (A \times B)$ and $\sum\...
2
votes
0
answers
78
views
value of $\frac{\sum_{k=0}^r{n\choose k}{n-2k\choose r-k}}{\sum_{k=r}^n {n\choose k}{2k\choose 2r} {(\frac{3}{4})}^{(n-k)}({\frac{1}{2}})^{2k-2r}}$ .
The question requires us to find the value of $\frac{\sum_{k=0}^r{n\choose k}{n-2k\choose r-k}}{\sum_{k=r}^n
{n\choose k}{2k\choose 2r} {\left(\frac{3}{4}\right)}^{(n-k)}\left({\frac{1}{2}}\right)^{...
2
votes
0
answers
60
views
Finding a formula for a sum that involves binomial coefficients
Is there a formula for this sum:
$$ \sum_{j=0}^k {n \choose j} {n \choose k-j} (-2)^j \left(-\frac13 \right)^{k-j} ?$$
It reminds me to Vandermonde's identity; but as you can see there is a slight ...
2
votes
0
answers
143
views
I've come up with two ways to evaluate $\sum_{1 \le j<k\le n} \frac{k}{k-j}$ but only one of them works
I've come up with two ways to solve this double sum but only one of them works:
$$ \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(...
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 ...
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_{...
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 $x\ge 0$...
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" ...
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 ...
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 ...
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". ...
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 < ...
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 $x$ ...
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 = {...
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 ...