All Questions
27
questions
2
votes
4
answers
273
views
How did Rudin change the order of the double sum $\sum_{n=0}^\infty c_n\sum_{m=0}^n\binom nma^{n-m}(x-a)^m$?
I see many people change the order of sum but I don't understand how they did that.
Is there is a way to change the order of the sum, $\sum\limits_{k=a}^n\sum\limits_{j=b}^m X_{j,k}$ and $\sum\...
1
vote
0
answers
92
views
Example of a specific polynomial
I need four non-trivial polynomials $P(x)$, $Q(y)$, $R(z)$ and $S(w)$ such that $$P(x)Q(y)R(z)S(w)=\sum_{i}a_i b_i c_i d_i x^i y^i z^i w^i+ \sum_{j\neq k\neq l\neq m}e_j f_k g_l h_m x^j y^k z^l w^m $$ ...
- 910
1
vote
1
answer
53
views
Summation limits with change of variables
I have the following sum (this is the context of QFT for period finding):
I am having trouble understaning the change in variables from line (1) to line (2).
I understand the sum over m gives ...
- 131
0
votes
0
answers
78
views
Double Summation over a Subset of a Cartesian Product
From the "Probability & Statistical Inference, 9th edition" by Hogg, Tannis, Zimmerman, it is stated that one of the properties of the Joint Probability Mass Function of Random Variables ...
- 123
0
votes
1
answer
398
views
Find a simple formula for $\sum_{k=1}^{n}(5k + 1)$
$$\sum_{k=1}^{n}(5k + 1) = \sum_{k=1}^{n}5k + \sum_{k=1}^{n}(1) = \sum_{k=1}^{n}5k + n = 5\frac{n(n+1)}{2} + n = 5n^2/2 + 5n/2 + n$$
Can something further be done ?
- 1,135
2
votes
3
answers
162
views
Evaluating $\sum \limits_{k=1}^{\infty} \frac{\binom{4k}{2k}}{k^2 16^k}$
I want to find the closed form of:
$\displaystyle \tag*{}\sum \limits_{k=1}^{\infty} \frac{\binom{4k}{2k}}{k^2 16^k}$
I tried to use the taylor expansion of $\frac{1}{\sqrt{1-x}}$ and $\frac{1}{\sqrt{...
0
votes
1
answer
39
views
Does anyone know how to bound decaying exponential series of the form $\exp(\sum_{k = i + 1}^n-C/(k+1))$
Consider a series with the form,
$$\exp(\sum_{k = i + 1}^n -C/(k+1))$$
where $C > 0$ is some constant and $n, i$ are integers with the assumption $n > i+1$
I wish to find some type upper-bound ...
- 557
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
54
views
Changing the index of the given sum
Calculate $f(i)$ from the following equation of sums:
$$\sum_{i=n^4}^{n^6}a_{(i+52)^7}=\sum_{i=n^3-70}^{n^9-70}a_{f(i)}$$
Using 1st and last limits of the sum we have, $f(n^3-70)=(n^4+52)^7$ and $f(n^...
- 772
0
votes
1
answer
25
views
Prove area using simple sums and given sum
I am supposed to prove that the given sum P has area h^3/3, I think I can do this by using the induction axiom but I'm pretty ...
- 3
2
votes
0
answers
38
views
Upper bound on $\frac{v_{1}}{1-c_{1}x}+\cdots+\frac{v_{n}}{1-c_{n}x}$
Let $v_{1},\ldots,v_{n}$ and $c_{1},\ldots,c_{n}$ be real numbers such that $v_{i}=2\alpha_{i}^{2}$ and $c_{i}=2\alpha_{i}$ for some $\alpha\ge 0$. My question is the following: Can I get, for $x\ge 0$...
- 638
2
votes
1
answer
160
views
How to prove that $ \Bigg(\sum^{n}\limits_{k=1}\sqrt{\frac{k-\sqrt{k^{2}-1}}{\sqrt{k(k+1)}}}\Bigg)^{2} \le n\sqrt{\frac{n}{n+1}}$ for $n\ge1$
I need to prove Prove the inequality
$$ \Bigg(\sum^{n}_{k=1}\sqrt{\frac{k-\sqrt{k^{2}-1}}{\sqrt{k(k+1)}}}\Bigg)^{2} \le n\sqrt{\frac{n}{n+1}}, $$
where $n$ is a positive integer.
Equivalently
$$\...
- 24.2k
0
votes
3
answers
2k
views
Finding the formula for the summation $\sum_{i=1}^n (2i-1) = 1+3+5+...+(2n-1)$.
I'm going through Calculus by Spivak and one of the questions is to find a formula for a summation.
$$\sum_{i=1}^n (2i-1) = 1+3+5+...+(2n-1).$$
I got the correct answer, $n^2$, but did it an ...
- 1,319
2
votes
1
answer
375
views
Generalization of Gauss's summation trick, relationship with proof of fundamental theorem of calculus?
Consider the following excerpt from Nets Katz's notes on calculus here, from pages 5 and 6.
As you can imagine, the argument generalizes to the sum of $k$th powers.$$S_k(n) = \sum_{j = 1}^n j^k.$$...
- 343
1
vote
2
answers
239
views
Formula for the general Cavalieri Sum: $S_n(p)=\sum\limits_{k=1}^{n} k^p\,\,\,n, p\in\mathbb N$ [duplicate]
What kind of formula is there that can be used for calculating the sum of power of $x$ of numbers from $1$ to $a$?
I know that the sum of numbers from $1$ to $a$ is $\ (n^2 + n)/ 2 \ $ and that the ...
- 111