All Questions
299
questions
3
votes
2
answers
533
views
Is knowing the Sum and Product of k different natural numbers enough to find them?
Can we uniquely identify the set of k different natural numbers (no two are the same) by knowing only their sum and product (and the value of k itself)?
- 133
3
votes
1
answer
923
views
Sum over subsets of $\{1,2,\ldots,n\}$ of terms involving a product over that subset
I'm attempting to perform a sum, using products, using all possible combinations, in a function.
How would I go about doing this? (I really need to find something that works.)
For example, say I ...
- 133
3
votes
2
answers
2k
views
How to go from a sum to a product and a product to a sum?
I have read here (third post down) that exponentials turn sums into products and logarithms turn products into sums. Can someone please further explain this?
- 5,630
3
votes
2
answers
178
views
Prove that $\frac1{a(1+b)}+\frac1{b(1+c)}+\frac1{c(1+a)}\ge\frac3{1+abc}$
I tried doing it with CS-Engel to get $$
\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{9}{a+b+c+ a b+b c+a c}
$$
I thought that maybe proof that $$
\frac{1}{a+b+c+a b+b c+a c} \geq \...
3
votes
4
answers
80
views
Sum and product of integers conserving order
I have the feeling this is true, but can't prove it:
$$\sum_n^An\lt\sum_n^Bn\implies\prod_n^An\lt\prod_n^Bn$$
Where $A\subset\mathbb N-\{0, 1\},B\subset\mathbb N-\{0, 1\}$
Example:
$$3+4\lt5+6\...
- 33
3
votes
2
answers
153
views
How to show this identity $\prod_{q=1}^k\frac{1}{1-qz}=\sum_{j=1}^{k}jz\prod_{q=1}^j\frac{1}{1-qz}+1$ avoiding a proof by induction
When looking at a nice problem regarding Stirling numbers of the second kind a challenge was to show the validity of
\begin{align*}
\color{blue}{\prod_{q=1}^k\frac{1}{1-qz}=\sum_{j=1}^kjz\prod_{q=1}^j\...
- 109k
3
votes
2
answers
81
views
$AM-GM$-ish inequality
Suppose $x_0, \cdots, x_n$ are positive reals. Suppose that:
$$\sum_{i = 0}^n \frac{1}{1+x_i} \leq 1$$
Then show that:
$$\prod_{i=0}^{n} x_i \geq n^{n+1} $$
I got to this problem by rewriting problem ...
- 4,284
3
votes
1
answer
146
views
Upper bound on product of finitely many non negative terms
Let $a_i, 1 \leq i \leq n$ be non-negative real numbers. Let $S$ denote their sum. To prove $$\prod_{k=1}^n(1+a_k) \leq 1 + \frac{S}{1}+\frac{S^2}{2!} + \ldots + \frac{S^n}{n!}$$
Let $S = \sum_{k=1}^...
- 4,500
3
votes
1
answer
1k
views
Triples of natural numbers with same sum and product
Im looking at pairs of triples of natural numbers without repititions such that the sums of the two triples are equal and the products of the two triples are equal.
To be precise: Let $x<y<z$ ...
- 132
3
votes
1
answer
197
views
Partition Proof
Let $\lambda$ be a partition of $N$ of rank $r$. How can I show that:
$$\sum_wx^\lambda(w)=f^\lambda(-1)^{t(\lambda)}\prod^r_{i=1}(\lambda_i-1)!(\lambda'_i-1)!$$
where $w$ ranges over all ...
user67253
3
votes
1
answer
307
views
Infinite Sum of Infinite Product
I've got an expression,
$$
\left( \prod^{n}_{i = 1} \mu a_i \right) \left(\sum^{n}_{j=1} \frac{1 - a_j}{\prod^{j}_{k=1} \mu a_k} \right)
$$
Where each $0 < a_j < 1 / \mu$ and $\mu > 1$, so ...
- 155
3
votes
1
answer
64
views
What is condition that the sum of $n$ complex numbers eaquals their product
Let $n\geq2$ and let $\{z_1,\dots,z_n\}$ be a set of complex numbers.
Is there a condition on the $z_i$'s such that
$$\sum_{i=1}^n z_i=\prod_{i=1}^n z_i$$
is identically true?
For $n=2$ the ...
- 1,008
3
votes
1
answer
383
views
Nested operation notation convention for evaluation (particularly for Pi and Sigma)
I have a question on the notation for two products.
For Pi:
Is this statement true?
$$\prod_{k=1}^{l}a_k\prod_{m=1}^{n}b_m = \prod_{k=1}^{l}\prod_{m=1}^{n}a_kb_m \neq a_1a_2\dots a_lb_1b_2\dots b_n = \...
- 255
3
votes
1
answer
92
views
Proving that, if $f(k)=\prod_{i=1}^ka_i+\sum_{b=1}^{k-1}(1-a_{k-b})\prod_{i=1}^ba_{k-b+i}$, then $f(k+1)=f(k)\cdot a_{k+1}+(1 - a_k)a_{k+1}$
Given that $$f(k) = \prod_{i=1}^k a_i + \sum_{b=1}^{k-1} (1-a_{k-b}) \prod_{i=1}^b a_{k-b+i}$$ for all $k$ where $(a_1, a_2, a_3, \ldots )$ are random constants, prove that: $$f(k+1) = f(k) \cdot a_{k+...
user851578
3
votes
0
answers
68
views
Evaluating a summation of product
Show that for any integer $k>1$
$$
\sum_{\substack{i_j \in \{0, 1\} \forall j < k, i_k = 0}} \prod_{j = 1}^{k} \left(i_j + (-1)^{i_j} \frac{a+ (j - 1) c - c \sum_{\lambda = 0}^{j - 1} i_\lambda}{...
- 51