All Questions
96
questions
1
vote
2
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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
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 ...
- 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
259
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
125
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
290
views
$\lim_{n\to \infty}(\lim_{x\to 0}( 1+\sum_{k=1}^n\sin^2(kx))^\frac{1}{n^3x^2} )$
$$\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n\sin^2(kx)\right)^\frac{1}{n^3x^2} \right)$$
$$=\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n(k^2x^2)\frac{\sin^2(kx)}{k^2x^2}\...
1
vote
0
answers
139
views
Solving a geometric-harmonic series
Find the value of $\displaystyle \frac21- \frac{2^3}{3^2}+ \frac{2^5}{5^2}- \frac{2^7}{7^2}+ \cdots$ till infinite terms.
found this problem while integrating $\arctan\left(x\right)/x$ from $0$ to $2$ ...
1
vote
1
answer
41
views
Iverson bracket - infinite additivity for pairwise disjoint sets
Suppose we have a sequence of mutually (pairwise) disjoint sets/events $B_1, B_2, B_3, ... $
EDIT: The sets $B_i$ ($i=1,2,3,\dots$) are subsets of some set $\Omega$.
For the Iverson bracket, is the ...
- 12.6k
3
votes
3
answers
168
views
find a closed form formula for $\sum_{k=1}^n \frac{1}{x_{2k}^2 - x_{2k-1}^2}$
Let $\{x\} = x-\lfloor x\rfloor$ be the fractional part of $x$. Order the (real) solutions to $\sqrt{\lfloor x\rfloor \lfloor x^3\rfloor} + \sqrt{\{x\}\{x^3\}} = x^2$ with $x\ge 1$ from smallest to ...
- 2,031
0
votes
1
answer
38
views
How to simplify: $C_{t} = r[a_{t} + \frac{1}{1+r} * \sum_{j=0}^{\infty} (\frac{1}{1+r})^j * E(w_{t+j})]$
I have to find C_t (Optimal Consumption for each period). Thank you!
$$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 : $$f(...
- 113
0
votes
1
answer
65
views
Calculating the final sum of an investment with a specific daily growth of rate over a period of time.
Calculating the final sum of an investment with a specific daily growth of rate over a period of time.
I do apologize if this question is very basic for the vast majority of people in this forum but ...
- 3
3
votes
1
answer
66
views
How can i simplify the following formula: $\sum\limits_{i,j=1}^{n}(t_{j}\land t_{i})$?
Consider the following time discretization $t_{0}=0< t_{1} < ... < t_{n} = T$ of $[0,T]$ where the time increments are equal in magnitude, i.e. $t_{j}-t_{j-1}=\delta$.
How can i simplify the ...
- 1,838
1
vote
1
answer
196
views
I wish to solve exactly this formula involving sums and products
I was solving a physics exercise and I encountered this formula:
$$\left< n_l \right>=\left[1+\sum_{k\neq l} \left(e^{bN(l-k)}\frac{\prod_{j\neq l} (1-e^{b(l-j)})}{\prod_{j\neq k} (1-e^{b(k-j)})}...
- 56
1
vote
1
answer
354
views
What is the fallacy in writing $x^2$ as the sum of $x$ $x's$? [duplicate]
It seems reasonable to write $x^2=x+x+...+x$ ($x$ times) but we run into a problem with derivatives if we do this.
The derivative of $x^2$ is $2x$ but the derivative of the sum on the right hand side ...
- 15
-1
votes
1
answer
105
views
How to cancel summation term that is multiplied by an equal summation
I have to demonstrate how to get from this initial equation:
$c = \frac{1}{1+R(1-\xi_t)}\cdot \frac{1}{N} \sum_{i=0}^{N-1}\left[ (1-\tau)w_t(i)n_t(i)+ \frac{1}{N} \left( \tau \sum_{i=0}^{N-1} \frac{w}{...
1
vote
1
answer
79
views
Simplify $\sum_{s=0: s \text{ even }}^\infty \sum_{m=0: \text{ even }}^\infty b_{s,m}x^s(1-x^2)^{\frac{m}{2}}$
I have the following double sum that I am trying to simplify into a single sum:
\begin{align}
\sum_{s=0}^\infty \sum_{m=0}^\infty b_{2s,2m}x^{2s}(1-x^2)^{m}
\end{align}
where $b_{s,m}$' are ...
- 2,941
3
votes
1
answer
317
views
Is there a closed form for $ \sum_{i=1}^n \left(\frac{i}{x + i}\right)^{i y} $?
I would like to find a closed formula for this equation:
$$
\sum_{i=1}^n \left(\frac{i}{x + i}\right)^{i y}
$$
Both the denominator and also the exponent is changing in each step. How is it possible ...
- 618
2
votes
3
answers
256
views
Evaluate $\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$
I would like to know how to evaluate $$\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$$
There are a couple of issues I have with this. Firstly, depending on the value of $x$, it seems, at ...
- 6,560
0
votes
2
answers
180
views
Sum of the roots of unity, $z_{1}^p+...+z_{n}^p$ [duplicate]
Let $z_1,...,z_n$ be the $n$ roots of unity. I am not able to find a value for the sum: $$z_{1}^p+...+z_{n}^p,\ p \in \Bbb N$$
I know that this sum can also be written as $$\sum_{k=0}^{n-1}e^{i(\frac{...
- 35
6
votes
3
answers
190
views
How to read and execute $\sum_{1 \leq \ell <m<n} \frac{1}{5^{\ell}3^{m}2^{n}}$
How to read and execute this sum?
$$\sum_{1 \leq \ell <m<n} \frac{1}{5^{\ell}3^{m}2^{n}}$$
I am having trouble to understand where is my error.
The question does not say, but I am assuming that ...
1
vote
2
answers
955
views
Leibniz formula for the nth derivative of $f(x)=x^{n-1} \ln x$
Problem : Calculate the derivative of the function $f: ]0,+\infty\left[\longrightarrow \mathbb{R}\right.$ defined by $f(x)=x^{n-1} \ln x$.
Solution
Let $g_{1}(x)=x^{n-1}$ et $g_{2}(x)=\ln x .$ So we ...
- 409
3
votes
3
answers
165
views
Evaluating double sum $\sum_{k = 1}^\infty \left( \frac{(-1)^{k - 1}}{k} \sum_{n = 0}^\infty \frac{1}{k \cdot 2^n + 5}\right)$
Find $$\sum_{k = 1}^\infty \left( \frac{(-1)^{k - 1}}{k} \sum_{n = 0}^\infty \frac{1}{k \cdot 2^n + 5}\right)$$
So far, I've gotten that the sum of the left is equal to $\log(2),$ meaning we have to ...
- 551
0
votes
1
answer
60
views
Alternating series estimation test proof
The first part of the proof of the error estimate theorem in integral calculus is confusing me. It states that $$\biggr\vert \sum_{n=0}^{\infty}(-1)^nb_n-\sum_{n=0}^N(-1)^nb_n\biggr\vert=\biggr\vert \...
- 2,072
0
votes
2
answers
49
views
Summation involving 2 variables
I am trying to understand how to expand a summation equation:
$$\sum_{j=1}^3 \sum_{i = j + 1}^4 (25-5i)$$
how do I expand the inner equation involving $i = j+1$ ?
Thanks!
- 1
0
votes
1
answer
40
views
given the sum of a finite sequence of real numbers $x_i$'s, find the $\sum_{i=1}^{N} e^{x_i}$
Let $\sum_{i=1}^{N} x_i $$=$$ 1 $ then what could one say about $\sum_{i=1}^{N} e^{x_i} $$=$$ ? $
- 159