2
votes
3answers
98 views

Double summation index problem

I often meet the following situation: $$\sum\limits_{n=0} ^\infty \sum\limits_{k=0} ^n f(k)g(n-k)=\sum\limits_{p=0} ^\infty \sum\limits_{q=0}^\infty f(p)g(q)$$ While intuitively this is very clear ...
1
vote
3answers
143 views

How do you derive the continuous analog of the discrete sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …$?

I was wondering what the rate of growth of the sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...$$ was, and found the related question, $n$th term of $1, 2, 2, 3, 3, 3, 4, 4 ,4, 4, 5...$, in which one of ...
1
vote
7answers
111 views

Error in proving of the formula the sum of squares

Given formula $$ \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} $$ And I tried to prove it in that way: $$ \sum_{k=1}^n (k^2)'=2\sum_{k=1}^n k=2(\frac{n(n+1)}{2})=n^2+n $$ $$ \int (n^2+n)\ \text d ...
1
vote
1answer
45 views

Permutations through different points

I'm watching Next (2007) and I'm trying to figure out a formula. The premise of the movie is that the protagonist can look into the future for two minutes and he is able to use this to alter his ...
1
vote
1answer
39 views

How to find the Direct Discrete Laplace Transform of ${2n \choose n}$

Some time ago I developed a discrete version of the Laplace transform for the purpose of calculating sums and solve finite difference equations with constant coefficients. The notes below are a ...
1
vote
2answers
54 views

Find a closed form for the generating function for this sequence

The sequence: $0, 0, 0, 1, 1, 1, 1, 1, 1, \ldots$ The book gives the answer of $\frac{x^3}{1-x}$ but I'm not sure how to get this answer. I understand the generating function of this sequence will be ...
2
votes
1answer
44 views

Particular solution of the recurrence equation $u_{n+2} + u_n = \sqrt{2}\cos[(n-1)\pi/4]$

I would like to solve the equation xx recurrence using the operator $E$, ie, $$ (E^2 + 1)u_n = \sqrt{2}\cos[(n-1)\pi/4] \quad \Rightarrow \quad u_n = \dfrac{1}{E^2 + 1}\{\sqrt{2}\cos[(n-1)\pi/4]\} $$ ...
1
vote
1answer
74 views

Expected value over many trials

I am a poker player and was talking to my friend about expected value. He claimed that if you play far enough above your bankroll, expected value can be negative, even if you have a skill edge. I ...
4
votes
5answers
322 views

Find a closed expression for a formula including summation

Let: $$\sum\limits_{k = 0}^n {k\left( {\matrix{ n \cr k \cr } } \right)} \cdot {4^{k - 1}} \cdot {3^{n - k}}$$ Find a closed formula (without summation). I think I should define this as a ...
1
vote
1answer
26 views

Discrete math: Sum of Geometric series on a problem - Notes make little sense.

I've been reading a PDF of slides from my Discrete Math I professor. The title is Sums, Products and Asymptotic Estimations. He gives us a problem to fire off the lecture, which is the following: ...
0
votes
2answers
29 views

Find the general solution to the following recurrence

Find the general solution to the following recurrence: $$nC_n=anC_{n-1}+bC_{n-1}$$ where a and b are constants.
0
votes
0answers
41 views

Simple Math Problem on Interval

It's not clear for me. I see this wikipedia page for a difference of half interval on $\mathbb{R}$ and interval on $\mathbb{R}$? For example $$ \{ (-\infty \le x \le a) \, \left|\, a \in \mathbb{R} ...
1
vote
1answer
46 views

I'm searching for the formula of the series $ \sum_{n=0}^{\infty}a^{n^l} $

I'm searching for the sum-formula (if exists) of the following power series: $$ \sum_{n=0}^{\infty}a^{n^l} $$ where $l=2,3,....$, and $|a|<1$.
2
votes
0answers
31 views

Computation of a 3-dimensional game matrix

For natural numbers $n_1 \leq n_2 \leq n_3$ we define $\beta(n_1,n_2,n_3)$ recursively to be the smallest natural number which is not among the numbers $\beta(m_1,m_2,m_3)$, where $m_1 \leq n_1 \leq ...
9
votes
3answers
301 views

Help with combinatorial proof of identity: $\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$

How to prove this identity? Can someone please give me some insight ? $$\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$$
3
votes
2answers
93 views

why generating function $A(z) = 1 + z + z^2 + \cdots$ can be denoted as $\frac{1}{1-z}$

It is easy to see that $1 + z + z^2 + \cdots$ is equal to $\frac{1}{1-z}$ when $1 > z > 0$ and for $z >= 1$, they are not equivalent. So I have thought $\frac{1}{1-z}$ is just a short for the ...
0
votes
1answer
29 views

A question on discrete sequences

Suppose for $1 \leq n \leq M$, we have a discrete sequence $a_n = (1 - 2^{n-M}) \gamma^n$, where $M$ is a fixed strictly positive integer, and $\gamma$ is a fixed strictly positive real number such ...
0
votes
1answer
19 views

Proving claims about sequences by induction?

I am learning how to prove claims about finite sequences right now. Can you help me prove or disprove the following claim? ...
0
votes
1answer
75 views

Induction Proof of: Find $f(n)$ when $n=2^k$, where $f$ satisfies the recurrence relation $f(n)=f(\frac{n}2)+1$ with $f(1)=1$

How can I proof this using regular induction and strong induction? $$f(2^k)=f(\frac{2^k}2)+1=f(2^{k-1})+1$$ $$f(2^{1-1})=f(2^0)=f(1)=1$$ $$f(2^{2-1})=f(2^1)=f(2)=f(1)+1=2$$ ...
1
vote
2answers
79 views

Get the Nth term of a sequence 1,2,4,7,13,24…

I have a sequence: 1,2,4,7,13,24,44,81, ... and I think it's like a Fibonacci sequence, however you add three number together and not two ("Tribonacci"?). So: $$ v_n = v_{n-1} + v_{n-2} + v_{n-3} $$ ...
0
votes
0answers
22 views

What is the general form of a finite sequence?

I originally used: $a_0,a_1,a_2,...,a_n$ where $n$ is an integer greater than or equal to 0 but I've realized that this is actually an infinite sequence because n can go to $\infty$. How can I bound ...
1
vote
1answer
31 views

How to find a pattern in this recursive sequence algorithmically?

I'm trying to find the closed-form of a sequence algorithmically. Here is the recursive sequence: $$w_k=w_{k-2}+k, \forall k \in \Bbb{Z} | k \geq 3, w_1=1, w_2=2$$ which produces this sequence: ...
1
vote
3answers
63 views

Finding the limit of a sequence by diagonalising a matrix

Consider the sequence described by: $\frac11 , \frac32 , \frac75 , ... ,\frac {a_{n}}{b_{n}}$ where $ a_{n+1} = a_n +2b_n $ and $b_{n+1} = a_n+b_n$ Find a matrix $A$ such that ...
3
votes
4answers
47 views

Is this a correct recursive sequence definition?

Take this definition: Is this definition of $s_k$ for $k\ge2$ correct? $s_k=6a_{k-1}-5a_{k-2}$ but where does the $a$ term come from? The book swaps $a$ and $s$ interchangeably.
0
votes
1answer
32 views

Number of configurations in a constrained nested loops and configuration back from serial

Consider 4 counters looping the digits 0, 1, 2 to form the various "configurations", like in : ...
0
votes
4answers
65 views

How to reduce this series to a single equation?

Somehow, my textbook was about to reduce this series to a single equation: I know that you can use the equation $$S=\frac{n(n+1)}{2}$$ for the sum of the first n integers but I don't think this ...
0
votes
2answers
25 views

Show $\sum_{k=2}^{2n+1}\frac{2}{k^2-1}= \frac{3}{2}-\frac{1}{2n+1}-\frac{1}{2n+2}$

Show that $$\sum_{k=2}^{2n+1}\frac{2}{k^2-1}= \frac{3}{2}-\frac{1}{2n+1}-\frac{1}{2n+2}$$ with the hint, "Write out the first six and last two terms. Then group them in pairs of two." Additionally, ...
0
votes
2answers
31 views

Help with proof by induction?

Prove $\frac{2(n-c)}{n+1} < 2$ where c is any natural So we assume $\frac{2(n-c)}{n+1} < 2$ is true, and so far I have $\frac{2(n+1-c)}{n+2} = \frac{2n-2c+2}{n+2} = \frac{2(n-c)}{n+2} + ...
0
votes
5answers
42 views

Recurrence problem for $a_5$

Assume that the sequence $\{a_0,a_1,a_2,\ldots\}$ satisfies the recurrence $a_{n+1} = a_n + 2a_{n−1}$. We know that $a_0 = 4$ and $a_2 = 13$. What is $a_5$?
0
votes
2answers
36 views

Compute $\sum_{i=0}^{2n} (-3)^i$ by splitting the series into two parts.

Compute $\sum_{i=0}^{2n} (-3)^i$ by splitting the series into two parts. How do I split it into two parts? All I can tell so far is that the sum is going to be a positive number (probably) because ...
-1
votes
2answers
24 views

Compute the sum $\sum_{i=0}^n 5^{i+1}-5^i$

Compute the sum: $$\sum_{i=0}^n 5^{i+1}-5^i$$ with the hint, "start by writing out (and expanding) the sum." So I did and got $$4 + 20 + 100...$$ with the appearance of going to infinity. Is ...
0
votes
1answer
23 views

How do you use induction on a recursive sequence using different variables?

I've been working on some recursive sequences for my Discrete class. I've understood most of them, but I've come to a new question which I'm confused about. A sequence $C_{0}$, $C_{1}$, $C_{2}$ is ...
0
votes
1answer
39 views

geometric series used to work out big O notation for resizing an array in a stack

It's a geometric series $$ 1 + 2 + 4 + \cdots + 2^k = \frac{1 - 2^{k+1}}{1 - 2} $$ Here, $2^k$ = N. You get $1 + 2 + 4 + \cdots + N = \frac{1 - 2N}{-1}$. Therefore, $2 + 4 + \cdots + N = 2N−2$. When ...
2
votes
2answers
112 views

Infinite Sum with Combination

I am trying to figure out what the following sum converges to: $$\sum_{n=0}^\infty {6+n\choose n}x^n(6+n),\qquad\qquad0<x<1$$ An answer would be great, but if you have an explanation, that'd ...
1
vote
2answers
70 views

Find a closed form for the sum $∑(x^3 - 2x)$ from $x=1$ to any number $n$

Find a closed form for the sum $∑(x^3 - 2x)$ from $x=1$ to any number $n$. Can someone explain to me what a closed form is and how to approach this problem?
0
votes
2answers
46 views

What's the maximum deviation from loan amortization

Suppose you have a loan with principle P and fixed interest rate i compounded daily. Suppose you make fixed payments every month, but not on the same day. The only constraint is that you make every ...
0
votes
1answer
39 views

$if An \subseteq A $ for all $n \in \mathbb{N}, $ then $ \bigcup_{n=1}^\infty An \subseteq A $

I was given this as an exercise in my discrete math class and I have been having a lot of trouble, I am not really sure how to approach a problem like this. Any help is appreciated!Thank you! (this is ...
1
vote
3answers
55 views

List naturals in ascending product order

Define an ascending product ordering as a sequence $(x_1,y_1), (x_2,y_2), \ldots$ with the following properties: Each pair of naturals is represented: For any integers $a\ge b>0$, we know that ...
0
votes
1answer
18 views

number of indices differences

I have a counting problem below: Let $n>2$ be integer and $p>0$ be a real number. For all $1\leq i<j\leq n$, suppose $a_{ij}$'s and $b_k$'s satisfy $a_{ij}=b_{j-i}=p^{j-i}$ ...
12
votes
3answers
303 views

$\frac{1}{7} + \frac{1\cdot 3}{7\cdot 9} + \frac{1\cdot 3\cdot 5}{7\cdot 9\cdot 11} + …$

If the sum $\frac{1}{7} + \frac{1\cdot 3}{7\cdot 9} + \frac{1\cdot 3\cdot 5}{7\cdot 9\cdot 11} + ...$ to 20 terms is $\frac{m}{n}$, reduced fraction, then what is $n-4m$? This is a question I dug ...
1
vote
1answer
48 views

Uniqueness proof of binary representation

I'm having trouble understanding this proof of uniqueness of binary representation of integers: Suppose there exists an integer $n$ with two different binary representations. Let these be: ...
1
vote
3answers
47 views

Help with sequence: $a_k = 5*3^k + 7*2^k$ - Induction

Let $a_k$ be a sequence, where $a_0 = 12$, $\;$ $a_2 = 29$ and $a_k = 5a_{k-1} - 6a_{k-2}$ , $k\geq 2$ . I need to prove, using induction, that $a_k = 5\times 3^k + 7\times 2^k \; , k\geq 0$ . ...
0
votes
0answers
23 views

Two dimention recursive recursive equation

I am unable to solve the following recursive equation which I must solve in my research problem. Please give me advice or solution to the problem. For $K=\min(N/2,C)$ and N,C T_c, T_s,p,T are ...
0
votes
0answers
32 views

Recurrence in Two Variables

Anyone know how to solve the following recurrence relation in two variables: $$ f(x,y) = b f(x-1,y) + c f(y,x-1) \\ f(x,0) = b^{(x-1)} \\ f(0,y) = 0 $$ (Note: repost of a post I asked yesterday with ...
1
vote
1answer
49 views

discrete mathematics , sequences, characteristic equation

Today in my discrete mathematics class we started combinatorics and also solved some some recurrence relation sequences using the characteristic equation. So my question is can you guys point me to ...
1
vote
1answer
53 views

Proving the form of a sequence's terms

How do I go about attacking this problem and what is it asking? Suppose that $\alpha^2 = \alpha + 1$ and suppose $F_n$ denotes the Fibonacci sequence. Show that $\alpha^3 = 2\alpha +1, \alpha^4 = ...
0
votes
0answers
24 views

String satisfying the condition

Given $N$, $A_0$, $B_0$, $L_0$, $A_1$, $B_1$ and $L_1$, find a sequence S consisting only of characters '$0$' and '$1$'(a total of N characters) such that: The number of '$0$'s in any consecutive ...
0
votes
1answer
25 views

function on a fixed length sequence of positive real number that induces lexicographic order

Let $S$ be the finite set of sequences of length $n$, whose entries are all real positive numbers. Can we define a function $f$ on $S$ such that the order $f$ induces on $S$ i.e $\le$ is the same as ...
0
votes
1answer
53 views

Base conversion using geometric series

I'm working on converting numbers in various bases and one question asks to convert $.2525...$ from decimal to octal. I know that the answer is $1/3$ and that it is necessary to use the infinite ...
1
vote
6answers
98 views

How do I start this summation (double series?)

I'm unfamiliar with this type of problem, but I've been asked to write out the result of n = 8. The problem: Prove that $$ \sum_{i = 1}^{n} \sum_{j = 1}^{i} f(i, j) = \sum_{j = 1}^{n} \sum_{i = ...