All Questions
96
questions
1
vote
2
answers
82
views
Upper rectangle area sum to approximate 1/x between $1\leq x\leq 3$
I am trying to figure out how to use rectangles to approximate the area under the curve $1/x$ on the interval $[1,3]$ using $n$ rectangle that covers the region under the curve as such.
Here is what I ...
- 1,071
1
vote
3
answers
66
views
I want to use integration for performing summation in Algebra
I am a class 9th student. Sorry if my problem is silly.
I am trying to find the sum of squares from 1 to 10. For this I tried summation, and it was fine.
But now I came to know that Integration can be ...
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
98
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
2
answers
170
views
Proof of weird formula for method of differences
Suppose we have to find sum of a sequence $t_1,t_2,t_3...t_n$.
For $1\le i\le n$, let $\triangle t_i=t_{i+1} -t_i$, $\triangle ^2t_i=\triangle t_{i+1}-\triangle t_i$ and so on ($\triangle ^{j}t_i=\...
- 433
0
votes
2
answers
115
views
Second-order partial derivative of $S = \sum_{k=0}^{2n}\left(\sum_{i=\max(0,k-n)}^{\min(n,k)}a_{i}a_{k-i}\right)$ [closed]
Given the following finite sum:
$S = \sum_{k=0}^{2n}\left(\sum_{i=\max(0,k-n)}^{\min(n,k)}a_{i}a_{k-i}\right)$
From this summation, I want to calculate explicitly each element $(i,j)$, i.e. the ...
- 85
0
votes
2
answers
251
views
Product of two finite sums $\left(\sum_{k=0}^{n}a_k\right) \left(\sum_{k=0}^{n}b_k\right)$
What is the product of the following summation with itself:
$$\left(\sum_{k=2}^{n}k(k-1)a_{k}t^{k-2} \right) \left(\sum_{k=2}^{n}k(k-1)a_{k}t^{k-2}\right) $$
Is the above equal to the double summation ...
- 85
0
votes
2
answers
124
views
Pi/product notation property applications problem
I have recently attempted to simplify this
$$ P(n) = \prod_{v=2}^{n} (2 + \frac{2}{v^2 - 1}) $$
I have reached an answer (which is wrong) through the following steps:
rearranging what is inside the ...
- 15
2
votes
2
answers
278
views
$\lim_{n\to \infty}(\lim_{x\to 0}( 1+\sum_{k=1}^n\sin^2(kx))^\frac{1}{n^3x^2} )$
$$\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n\sin^2(kx)\right)^\frac{1}{n^3x^2} \right)$$
$$=\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n(k^2x^2)\frac{\sin^2(kx)}{k^2x^2}\...
1
vote
0
answers
138
views
Solving a geometric-harmonic series
Find the value of $\displaystyle \frac21- \frac{2^3}{3^2}+ \frac{2^5}{5^2}- \frac{2^7}{7^2}+ \cdots$ till infinite terms.
found this problem while integrating $\arctan\left(x\right)/x$ from $0$ to $2$ ...
1
vote
1
answer
41
views
Iverson bracket - infinite additivity for pairwise disjoint sets
Suppose we have a sequence of mutually (pairwise) disjoint sets/events $B_1, B_2, B_3, ... $
EDIT: The sets $B_i$ ($i=1,2,3,\dots$) are subsets of some set $\Omega$.
For the Iverson bracket, is the ...
- 12.7k
3
votes
3
answers
167
views
find a closed form formula for $\sum_{k=1}^n \frac{1}{x_{2k}^2 - x_{2k-1}^2}$
Let $\{x\} = x-\lfloor x\rfloor$ be the fractional part of $x$. Order the (real) solutions to $\sqrt{\lfloor x\rfloor \lfloor x^3\rfloor} + \sqrt{\{x\}\{x^3\}} = x^2$ with $x\ge 1$ from smallest to ...
- 2,031
0
votes
1
answer
38
views
How to simplify: $C_{t} = r[a_{t} + \frac{1}{1+r} * \sum_{j=0}^{\infty} (\frac{1}{1+r})^j * E(w_{t+j})]$
I have to find C_t (Optimal Consumption for each period). Thank you!
$$C_{t} = r[a_{t} + \frac{1}{1+r} * \sum_{j=0}^{\infty} (\frac{1}{1+r})^j * E(w_{t+j})]$$
Where,
$$w_{t+j} = \begin{cases} w + \...
1
vote
1
answer
109
views
How to factorize and solve equations with $\Sigma$ notation?
I have a few doubts about the properties of sigma notation, $\Sigma$ . My questions rely on factorization and solving equations with $\Sigma$.On account of the fact that my questions are correlated, I ...
user1056027
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