All Questions
20
questions
17
votes
7
answers
8k
views
Chain rule for discrete/finite calculus
In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in ...
- 1,787
11
votes
9
answers
6k
views
How to compute the formula $\sum \limits_{r=1}^d r \cdot 2^r$?
Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$
how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$
Thank you
- 223
8
votes
1
answer
251
views
Changing Variables in Discrete Calculus
In discrete calculus one soon meets the $h$-difference operator $$\Delta_h[f(x)] = f(x+h) - f(x)$$
and we often define $\Delta = \Delta_1.$ We can similarly define the indefinite sums $\Delta_h^{-1}$ ...
- 153
5
votes
1
answer
58
views
How to evaluate $\sum\limits_{n \geq 0} \left(S_{n + 2} + S_{n + 1}\right)^2(-1)^n$, given the multivariable recurrence relation?
The given multivariable recurrence relation is that for every $n \geq 1$
$$S_{n + 1} = T_n - S_n$$
where $S_1 = \dfrac{3}{5}$ and $T_1 = 1$. Both $T_n$ and $S_n$ depend on the following condition
$$
\...
- 3,364
4
votes
1
answer
690
views
Formula for nth derivative of partial sum of geometric series.
I am trying to find a formula for either
(1) the $n$th derivative for the following $m$th partial sum:
$$\frac{d^n}{dx^n} \sum_{i=0}^m x^i$$
or (2) the $n$th derivative of the infinite series given by
...
- 6,099
3
votes
3
answers
138
views
Infinite Sum with Combinatoric Expression
Calculate the value of the infinite sum, where $0<p<1$ and $r,n\in\mathbb{Z}_{\ge1}$:
$$\sum_{y=1}^\infty \binom{r+y-1}{y}y^n(1-p)^y$$
Just for context, this is the $n^{th}$ moment of the ...
- 2,365
3
votes
5
answers
5k
views
Deriving the summation formula for $x^2, x^3,\ldots,x^n$
How is the summation formula's for $x,x^2,x^3,x^4,\ldots$ derived? I know how to do it for $x$ which is $n^2/2 + n/2$ but I am having hard time deriving the summation formula for $x^n$ on my own. I ...
- 835
2
votes
1
answer
106
views
Function satisfying $f(z) = 1 + z f\left(\frac{z}{1+z}\right)$
Iterate the following equation to obtain an explicit formula for $f( z)$:
$$
\begin{align*}
f( z) = 1 + z f\left( \frac{z}{1 + z}\right)
.\end{align*}
$$
Iterating this equation one obtains
$$
\begin{...
- 575
2
votes
3
answers
118
views
Sum with two indices, how to handle the condition $i\neq j$?
I'm given
$$
\sum_{i,j=1, i \neq j}^{N} ij \tag 1
$$
I assume this is a short hand for
$$
\sum_{i=1}^{N}
\Bigg (
\sum_{j=1}^{N} ij
\Bigg )
\qquad? \tag 2
$$
But where does $i\neq j$ belong, to the ...
- 2,362
2
votes
1
answer
73
views
Integration of 1 to n...
Anybody can explain how, summation of $1$ to $N$, can be replaced with integration and result leads to $(1/2)N^2$:
$$
\sum_{i=1}^N i \sim \int_1^N x \ \mathrm{d}x \sim \frac{1}{2}N^2
$$
Note: Image ...
- 131
2
votes
2
answers
178
views
Evaluate the Finite Sum with Binomial Coefficient [duplicate]
Evaluate the following sum:
$$\sum_{k=0}^n\binom{n}{k}\frac{k}{n}x^k(1-x)^{n-k}$$
This almost looks like the Binomial Theorem, except there's that pesky little $\frac{k}{n}$ term in there that ...
- 2,365
1
vote
4
answers
957
views
Summation of n-squared, cubed, etc. [duplicate]
How do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great.
- 493
1
vote
3
answers
418
views
$T(n)=2T(n-1)+n,T(1)=1,n\ge2$ [duplicate]
Solve the recurrence $T(n)=2T(n-1)+n,T(1)=1,n\ge2$
My approach:
$$T(n)=2T(n-1)+n$$
$$T(n)=2(2T(n-2)+(n-1))+n$$
$$T(n)=4T(n-2)+2(n-1)+n\dots$$
$$T(n)=2^kT(n-k)+2^{k-1}(n-k+1)+\dots+2^{k-k}n$$
From ...
- 2,540
1
vote
1
answer
38
views
Prove Value of Summation with Combinatoric
Prove that:
$$\sum_{k=0}^n\binom{n}{k}\left(\frac{k}{n}\right)^2x^k(1-x)^{n-k}=\frac{x(1+x(n-1))}{n}$$
I've managed to simplify the sum up until this point:
$$\frac{\sum_{k=1}^n\binom{n-1}{k-1}kx^k(...
- 2,365
1
vote
1
answer
1k
views
Find generating function for the sequence 0, 0, 0, 0, 3, 4, 5, 6, ...
1.Derive the generating function for the sequence $$0, 0, 0, 0, 3, 4, 5, 6, . . .$$
2.Derive the generating function for the sequence $$0, 0, −12, 36, −108, 324, .. .$$
So the first function ...
- 33