1
vote
1answer
49 views

Infinite series for recurrence

Question 1 If I define $A(z) = \sum_{n \ge 0} a_n \frac{z^n}{n!} \tag 1$ (where $a_n$ are $3\times 3$ constant matrices indexed with n), then can we re-write $\sum_{n \ge 1} a_{n-1} \frac{z^n}{n!} ...
7
votes
3answers
241 views

Solving recurrence relation: Product form

Please help in finding the solution of this recursion. $$f(n)=\frac{f(n-1) \cdot f(n-2)}{n},$$ where $ f(1)=1$ and $f(2)=2$.
3
votes
1answer
76 views

Limit of a recursiv trigonometric function

So while doing my math homework I recently stumbled upon this little thing: It seems that $y_n = \sin(y_{n-1}) + k$ with arbitrary $y_0$ and $k$ converges to a definite value for $n \to \infty $. ...
3
votes
1answer
39 views

$n$th term of a recursive formula

I have a formula $$ 1 + px + \dfrac{p(p-1)}{1*2}x^2 + \dfrac{p(p-1)(p-2)}{1*2*3} x^3 $$ can someone please tell me what the formula is for the $n$th term of this recursive definition ? Do I have to do ...
0
votes
2answers
86 views

Repertoire Method. How to know when equations are valid

I'm trying to work a few examples from the Repertoire Method from the Book Concrete Mathematics. I'm working through the following recurrence: $f(0) = 1$ $f(n) = 2f(n-1) + n$ Which I generalized ...
0
votes
1answer
59 views

how do you rewrite a recursive formula to find its roots

Let $(x_n)$ be the sequence defined by $x_1=2$ and the recursive formula $x_{n+1} = \frac12 + \sqrt{x_n}$. Rewrite the recursive formula in the form $$ x_n - x_{n+1} = ax_{n+1}^2 + bx_{n+1} + c$$ ...
0
votes
1answer
56 views

How to solve recursive function

I've recently been doing some limits with circuits and such, and I came up with the following equation, $R$ being a constant: $$f(x) = \frac{f(x-1)*R}{R+f(x-1)}+R$$ with $f(1)=2$. I know that this ...
0
votes
1answer
35 views

Find another recursive algorithm that is equal to a series

I have the following sequence: $$ y_n = \int_0^1 \frac{x^n}{x+5}\,dx, n = 0,1,\dots $$ Now I have the following recursive algorithm which is equal to the sequence: $$ y_0 = \log{6} - \log{5} $$ $$ ...
0
votes
2answers
88 views

How to prove that a series is equal to a recursive algorithm

I have the following sequence: $$ y_n = \int_0^1 \frac{x^n}{x+5}\,dx, n = 0,1,\dots $$ Now I have the following recursive algorithm: $$ y_0 = \log{6} - \log{5} $$ $$ y_n = \frac{1}{n} - 5y_{n-1}, n ...
1
vote
1answer
56 views

Using series to produce guess for algorithm analysis

I need to find the upper asymptotic bound for the recursion: $$ T(k) = 2T(k-1)+\frac{1}{k} $$ I was able to determine: The height of this tree is $k-1$. The cost of each level is ...
3
votes
1answer
200 views

Computing sums of divisors in $O(\sqrt n)$ time?

I have a sequence: $1,3,5,8,10,14,16,20,23,27,\dots$ I know that the recursive relation is: $$p[i] := p[i-1] + \text{number of factors of $i$}, \quad \text{with $p[1]=1$.}$$ How do I solve this ...
0
votes
4answers
586 views

Second-Order, Linear Inhomogeneous Recurrence Relation With Constant Coefficients

How does one solve the general recurrence relation $$s_n=\alpha s_{n-1}+\beta s_{n-2}+\zeta(n)?$$
0
votes
1answer
200 views

Recursive Function for Pyramid-Scheme

consider a group which has 1 user. each month, every user can bring another user to join the group. the user that has been joined for 3 months, should leave the group. calculate the total membership ...
0
votes
2answers
61 views

Equation of a curve whose difference in ordinate values form an arithmetic sequence

I have the following recurrence equation that I have obtained while trying to solve a problem:- $$T(0) = 1$$ $$T(n) = T(n-1) + 9n - 8: n \ge 1$$ The values of $T(n)$ for $n = 0,1,2,... $ are as ...
1
vote
0answers
63 views

How to deduce the results of response time by this trajectory approach?

First, we denote this: And And we get this right property( $last_i$ means the last node on $τ_i$): And: $Smin_i^h$ = $\sum_{h'=first_i}^{h-1} ({C_i^{h'} + L_{max})}$ $Smax_i^h$ = ...
2
votes
2answers
417 views

Convergence of a Recursive Sequence - An Example

Consider the sequence $\displaystyle x(k+1) = \frac{1}{2}\left(x(k) + \frac{a}{x(k)}\right)$ where $x(k)$ stands for the $k$th term of the sequence. What does this process converge to, and what is ...
2
votes
0answers
117 views

Recursive Sequence from Finite Sequences

I'm searching for the name of these kind of sequences: Suppose you start off with a finite sequence containing one term: S0 = {3} To get the next sequence you ...
0
votes
1answer
83 views

Bounded recursive sequence

I would like to know if there are known bounded recursive sequence (non monotonic): It shouldn't be a constant, neither a convergent sequence, nor a periodic one. (I am not asking for a true random ...
4
votes
2answers
262 views

Sum of the series formula

I need to figure out the sum of the series as quickly as possible in a program given n and k: $$f(n,k)= ...
1
vote
1answer
239 views

Derive a General formula for each term of this periodic sequence?

I have a sequence $a_0 = 1, a_1, a_2, a_3, \dots$ such that $a_4 = a_{24}$ which implies that period repeats after $a_{24}$ to $a_{43}$. Each $a_n$ depends on $a_{n-1}$ only. I need general term for ...
0
votes
6answers
295 views

Obtain the formula for the following sequence

I can't seem to figure out how to find an algebraic formula for the following sequence of numbers. $$0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0$$ Can somebody ...
0
votes
1answer
3k views

Convert Recursive to Closed Formula

I got a particular sequence defined by the following recursive function: $$T_n = T_{n-1} \times 2 - T_{n-10}$$ I need help converting it to a closed form so I can calculate very large values of n ...
0
votes
3answers
496 views

Recursive sequence - Convergency and Limit

i just started university so im pretty new to all this new math. My problem is to solve this recursive sequence: $a_{n+1} = a_{n}^3$ with: $a_{0} = \frac{1}{2}$ ...
1
vote
2answers
476 views

The first four terms x, y, z, w of an arithmetic sequence

I have been attempting this question for the past 3 days with no luck: ...
2
votes
1answer
69 views

Incomplete “round trip” of taking a minimum, then a maximum, from a positively skewed distribution

Let's say you have a distribution that is either symmetric or positively skewed (and defined over 0-1). Call it F. Then, you find the distribution of the minimum of n>1 draws from F. Call it Fmin. ...
0
votes
1answer
136 views

Can I write this series as a recursive function

I've been given the task of computing two series programmatically one is of the form: $ 1 + \frac{1}{2} +\frac{1}{3}+\frac{1}{4}...+ \frac{1}{n} $ The other one is an estimation for $\pi$ which is ...