All Questions
Tagged with summation algebra-precalculus
89
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
302
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:
$$\sum_{k=0}^{n-1}\cos (a+k d) =\frac{\sin( \frac{nd}{2})}{\sin ( \frac{d}{2} ...
- 6,550
4
votes
2
answers
78
views
Is $\left(\sum_{n=1}^N\frac{a_n}{N}\right)^N\left(\sum_{n=N+1}^{2N}\frac{a_n}{N}\right)^N≠\left(\sum_{n=1}^{2N}\frac{a_n}{2N}\right)^{2N}$?
Let
$$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_{...
- 3,405
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(\...
- 607
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 ...
- 4,845
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
175
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}
$$
Is there a way of simplifying this expression?
For example, provided $\{x_i\}$ is bounded and "smooth" enough ...
- 3,405
2
votes
0
answers
39
views
Simplifying $\sum_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$ in two ways gives different results
I want to calculate the result of
$$\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