All Questions
299
questions
4
votes
1
answer
7k
views
Can Pi prod be expressed using Sigma Notation?
Can $\prod(x)$ be expressed in terms of $\sum (x)$?
- 6,834
4
votes
2
answers
82
views
How to define this pattern as $f(n)$
Given a binary table with n bits as follows:
$$\begin{array}{cccc|l}
2^{n-1}...&2^2&2^1&2^0&row\\ \hline \\ &0&0&0&1 \\ &0&0&1&2 \\ &0&1&0&...
- 265
4
votes
1
answer
88
views
Formulating an alternating sum of product combinations
Consider some list $A=(a_1,a_2,\cdots,a_n)$. I'd like to find a closed form for the following operation.
$$f(A)=\sum_{k=1}^n(-1)^{k-1}s_k= s_1-s_2+\cdots(-1)^{n-1}s_n.$$
Where $s_k$ is the sum of all ...
- 4,472
4
votes
1
answer
115
views
How can I evaluate the below mentioned series without using a computation software?
I have been trying to evaluate $\displaystyle\sum_{m=0}^{2^{2^5}-1}\frac{2}{\prod_{n=1}^5\bigl((m+2)^{\frac{2}{n}}+(m)^{\frac{2}{n}}\bigr)}$ for quite a long time. I tried various approaches but ...
- 121
4
votes
1
answer
1k
views
Sum and Product Puzzle and Prime Factors
Suppose we have two number $X$ and $Y,$ such that $1 < X < Y < 100,$ and $X + Y ≤ 100.$ Sue is given $S = X + Y$ and Pete is given $P = XY.$ They then have the following conversation:
Pete: '...
- 143
4
votes
2
answers
114
views
Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$
In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum:
$$ \underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( \sum_{c=0}^{b+...
- 356
4
votes
1
answer
3k
views
Proof of Inequality involving sum and product without induction
How could you prove these inequalities wihout induction:($a_k$ are non-negative)
1)$\prod_{k=1}^n(1+a_k)\ge1+\sum_{k=1}^n a_k$
2)$\prod_{k=1}^n(1+a_k)\le1+\frac{\sum_{k=1}^na_k}{1!}+\ldots+\frac{(\...
- 7,085
4
votes
1
answer
87
views
Simple formula for the $n$-ary version of $(x,y) \mapsto \frac{x+y}{1-xy}$
Let $x * y = \frac{x + y}{1 - xy}$. I want a single formula for $x_1 * x_2 * \ldots * x_n$, for all natural $n$.
In order to generate plausible candidates, let's see what happens at small values of $...
- 1,755
4
votes
2
answers
1k
views
Summation and Product Bounds
If I have a sum or product whose upper index is less than its start index, how is this interpreted? For example:
$$\sum_{k=2}^0a_k,\qquad \prod_{k=3}^1b_k$$
I want to say that they are equal to the ...
- 17.8k
4
votes
1
answer
406
views
How do you prove by induction when summation and product notation are involved?
Firstly, how would you solve an equation such as the Binomial Theorem by way of Mathematical Induction, and how could you use that to prove the following?
$$\left(1 + \frac1n\right)^n = 1 +\sum_{k=1}^...
user537153
4
votes
0
answers
127
views
Products of trig functions and the Thue–Morse sequence
I was studying transformations of finite products of trig functions into sums, and empirically observed that the following curious identity appears to hold for all non-negative integer $m$:
$$\prod_{...
- 47.3k
4
votes
0
answers
96
views
Index of summation for a product
In the expression
$$\sum_{k=1}^nk=\frac12n(n+1)$$
$k$ is called the index of summation. What is $j$ called in
$$\prod_{j=1}^nj=n!$$
I have seen web pages like https://math.illinoisstate.edu/day/...
- 16k
4
votes
3
answers
1k
views
Number equal to the sum of digits + product of digits)
Are every number equal to (sum of digits + product of digits) in a given base only two digits long ?
Thought about limiting like this :
$$b^{(n - 1)} \leq N = \text{Product digits} + \text{Sum ...
- 41
3
votes
2
answers
825
views
Sum of real numbers that multiply to 1
I've seen a question in my math book with this explanation above it: "If the product of n positive real numbers is 1, then the sum of these numbers must be more than n". I was wondering if this is ...
- 219
3
votes
4
answers
339
views
How find this value $\prod_{k=1}^{\infty}\left(1+\frac{1}{k^5}\right)$
Find the value
$$\prod_{k=1}^{\infty}\left(1+\dfrac{1}{k^5}\right)$$
I know this :How find this $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^6}\right)$
and maybe can find the $2k+1$? can you someone konw ...
- 93.6k