1
vote
1answer
31 views

Asymptotics of logarithm: $\frac{1}{n}\ln(a+o(1)) = \frac{1}{n}\ln(a)+o(\frac{1}{n})$

I am having problems with the use of the little oh notation my professor is adopting in the solutions to some exercises. As an example I do not understand why $$ \frac{1}{n}\ln(a+o(1)) = ...
0
votes
1answer
31 views

What notation to use for a sequence of integers that end with digit 5?

I need to solve a low high school home work and I ask a question about the most correct notation. The problem is to build a set of circles with $r$ and $d$ such that $d=5, 15, 25, 35,...d_{+_1}$ and ...
4
votes
4answers
761 views

Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
1
vote
1answer
84 views

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$?

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$? I am reading my textbook on analysis, and it seems to use 'sequence of functions' to describe both ...
1
vote
1answer
32 views

Representing a for loop with modulus in formal notation

I have the following section of code I am writing for research. Basically I need to formally represent the mathematical notation behind a set that follows: ...
3
votes
2answers
31 views

Big O-notation and series

While there are many questions regarding Big O-notation and in particular, its usage when it comes to series, none fit my question perfectly: possibly because I am so unfamiliar with the notation. ...
2
votes
1answer
92 views

Which is the best notation for a sequence?

In a set, the order of its elements is (as far as I know) not important; in a sequence, the order of its elements is important. Which is the notation I should use in order to define a sequence? I ...
3
votes
1answer
89 views

$f_{n+1}(x)=f_n(x+1)-f_n(x)$ functional equation and “classification of functions”

Doing a quiz I found a question of this kind "given $a_0, a_1, a_2, ...,a_n$ find $a_{n+1}$" In order to find the $f$ such that $f(a_n)=a_{n+1}$ I tryed for a function like $f(x)=k+x$ ...
0
votes
0answers
37 views

Precise notation of a sum of a sequence

I need help in rewriting the support of a function $f$ in a more compact or precise way given its upper bound $b$ and lower bound $a$ as \begin{eqnarray} b&=&\max\left( \sum_{n}\alpha_n ...
1
vote
2answers
46 views

How to represent this sequence mathematically?

I need to represent the sequence of pairs $$(N,0), (N-1,1), (N-2,2), \ldots , \left( \frac{N}{2}, \frac{N}{2}\right) $$ in a way I can use in a formula. Is there any way to do this? Thanks!
2
votes
2answers
85 views

Taylor series of $\sqrt{1+x}$ using sigma notation

I want help in writing Taylor series of $\sqrt{1+x}$ using sigma notation I got till $1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\ldots$ and so on. But I don't know what will come in ...
2
votes
3answers
166 views

Equation with the big O notation

How I can prove equality below? $$ \frac{1}{1 + O(n^{-1})} = 1 + O({n^{-1}}), $$ where $n \in \mathbb{N}$ and we are considering situation when $n \to \infty$. It is clearly that it is true. But I ...
0
votes
2answers
52 views

Sequence Notation in Analysis

If real sequence is a function from the set $\mathbb N$ to the set $\mathbb R$ and function is represented by $(a,b)$, where $a$ is domain and $b$ is range, then why do we represent sequence only by ...
0
votes
2answers
62 views

Is there a formal name for $(x+1)^3 - x^3 = 3x^2 + 3x + 1$

$(x+1)^3 - x^3$ can be simplified to $3x^2 + 3x + 1$. Is there a formal name for this(some theorem)?? Also does this have a relation to geometric progressions since $3x^2, 3x, 1$ is a geometric ...
0
votes
0answers
27 views

How to define a conditional length for a sequence?

Let $\langle a_i \rangle_{i=1}^{n}$ be a sequence from $i$ to $n$. And let $x$ be a value that can vary, example: $x=100$. Which would be the best notation if I wanted to put a condition to limit the ...
0
votes
1answer
45 views

Sequence with a fixed last element Notation

I was trying to write a sequence of two different elements (that always appear in order) with a fixed last element, for an example: $A_1, B_1, A_2, B_2, A_3, B_3, A_4$. I'm not sure which would be the ...
0
votes
0answers
51 views

Correct notation in this case

I want to define $y_n:=(1,\frac{1}{2},...,\frac{1}{n},0,..)$(hence, a sequence) for all $n \in \mathbb{N}$. And I was wondering whether $y_n:= \sum_{i=1}^{n} \frac{1}{i} (\delta_{ip})_{p \in ...
1
vote
1answer
136 views

What does $(X_n, Y_n)$ mean? ($X_n, Y_n$ are two sequences of real numbers)

I apologize for this very basic definition question, but I am stumped because my professor uses this notation without any introduction. Verbatim, "Recall that one of the properties of algebraic ...
0
votes
2answers
155 views

Reference a single element within a set

Is there a notation to reference a single element within a set? Let's say I have a set n = {1, 2, 4, 8, 16}. If I wanted to use a single element from this set, is ...
1
vote
2answers
80 views

How to describe a n-tuple of sequences

When I write computer programs, I often use something called a multidimensional array. I think the concept would be equivalent to an n-tuple of finite sequences. Suppose I have the following four ...
0
votes
1answer
40 views

Notation to describe the adding of a constant to all terms of a sequence

I've been struggling to get down the proper mathematical notation for sequences. Suppose I have the following sequence: $$A = (4, 3, 7, 3, 1)$$ How do I describe the addition of a constant to all ...
0
votes
1answer
64 views

Joining finite sequences [duplicate]

How do I describe the joining of two finite sequences in mathematical notation? For example, suppose the following: $$ A=(a_i)_{i=1,2}=(4,2)\\ B=(b_i)_{i=1,2}=(9,5)\\ C=(c_i)_{i=1,...,4}=(4,2,9,5) $$ ...
0
votes
2answers
133 views

I don't understand this notation… - Series with ln

I found this notation in my book $$ \sum\limits_{i=1}^n \ln^n3 $$ and I don't know how to interpret it. Is it $$ \sum\limits_{i=1}^n \ln((1^n)\cdot3)\;? $$ And btw, how to check if this series ...
0
votes
0answers
137 views

Time series notation

I'm developing formal software requirements specifications for processing time series data and thus need to mathematically describe time series and operations on time series. Is there establish ...
0
votes
2answers
27 views

Notation for number of value changes in a sequence

Let $A=\{a_{1}, a_{2}, a_{3}, a_{4}, ...,a_{n}\}$ be a finite sequence , where $a \in \mathbb{N}$. I would like to know the notation for something similar to a change rate. If I programmed, what I ...
0
votes
1answer
40 views

Mathematical description of an identifier which consists of text and numbers [closed]

in my code I use an artificial identifier of the form UIxxx where xxx is in the range of 1 to 999. In my assay, where I describe this unique identifier I am not sure how to mathematically describe it. ...
4
votes
1answer
615 views

Is there a common symbol for concatenating two (finite) sequences?

Say we have two finite sequences $X = (x_0,...,x_n)$ and $Y = (y_0,...,y_n)$. Is there a more or less common notation for the concatenation of these sequences, like $\sum (X,Y) = ...
0
votes
2answers
89 views

Geometric Sequence

What does the parentheses mean in this problem? $$\sum_{K=0}^4 {4\choose K} x^k y^{4-k}$$ ...
1
vote
1answer
39 views

What is $\phi(k)$ in $\sum_{k=1..n} \phi(k)\lfloor n/k \rfloor^2$?

I would like to compute a general $n$-term of a sequence $$ 1, 5, 12, 24, 37, 61, 80, \dots$$ However I do not understand what $\phi$ refers to in the formula at http://oeis.org/A018806: ...
2
votes
1answer
62 views

is $\frac{\{1, \ldots ,n\}}{n+1}$ proper notation when $n\geq1$?

I am trying to explain R code: (1:n)/(n+1) such that: > n <- 4 > (1:n)/(n+1) [1] 0.2 0.4 0.6 0.8 I might use ...
14
votes
4answers
733 views

Donald Knuth's summation notation confuses me.

I do not understand a lot of cases of Knuth's summation notation in Concrete Mathematics. In general, I cannot seem to get a grasp on the commutative law as applied to manipulating sums. The ...
4
votes
3answers
294 views

Is it meaningful that sequence or expression has limit $\infty$?

I have recently started my studies in university (mathematics), and we're now studying sequences. I was surprised when the professor wrote that: $$\lim_{n \to \infty} a_n=\lim_{n \to ...
2
votes
2answers
6k views

What does ! mean in sequences?

I'm doing a sequences problem where I have to write the first five terms of a sequence. It looks normal, but there is an exclamation mark on the denominator: $$a_n = \frac{1}{(n + 1)!}$$ & ...
0
votes
1answer
402 views

Represent loop nests as multiple summations?

This may be trivial but I would appreciate if someone could point me in the right direction here.. I am trying to express the number of instances in a loop nest in a general form. As a mathematical ...
2
votes
2answers
981 views

Convert for-loop into mathematical expression

I'm facing a simple problem actually: for (int i = 0; i < noutput_items; i++){ out[i] = in[i] * in[i]; } When I want to formalize this as a math ...
1
vote
1answer
480 views

Sequence Notation

I have come across the following in the book "Principles of Program Analysis" by Nielson, Nielson and Hankin to represent a sequence and I am unsure of what its constituent parts mean. Obviously ...
3
votes
2answers
796 views

Big O notation, $1/(1-x)$ series

I am doing calculus and I find this notation extremely confusing even after reading textbook and notes. Here is a question I was trying to do and I think I cannot do this yet because I dont fully ...
2
votes
4answers
1k views

What is the summation operator used for in Mathematics?

I am quite experienced in programming and I know that summation in math is similar to a for loop: It runs the specified operations for $x$ amount of times. What I want to know is how this is useful in ...
0
votes
1answer
177 views

Eventually always notation

Let $X$ be a set and $A\subset X$. For a sequence $F = (f_n)_{n\geq 0}$ of elements of $X$ we say that $F$ is eventually always in $A$ if for some $N\geq0$ it holds that $f_n\in A$ for all $n\geq N$. ...
1
vote
0answers
685 views

Notation for a subsequence of a sequence

If we have a sequence (an ordered list) $$ S=(s_0,s_1,...,s_n). $$ What is the notation for expressing that $S'$ is a (ordered) subsequence of $S$?
2
votes
2answers
99 views

Elementary question on Sums notation

I reading on Sums and I am reading about the difference between using a generalized Sigma notation and the delimited form. Ok, I understand that the generalized form is more expressive. But I ...
1
vote
1answer
441 views

Little $o$ notation and series

I have this question: Consider the series $e^{\tan(x)} = 1 + x + \dfrac{x^{2}}{2!} + \dfrac{3x^{3}}{3!} + \dfrac{9x^{4}}{4!} + \ldots $ Retaining three terms in the series, estimate the ...
5
votes
2answers
997 views

Formally writing about lists (tuples), and notation analogous to set notation

Is there any formal notation for dealing with lists, rather than sets? e.g. if I have a set $X=\{x_1,\dots,x_n\}$ and I want to add a new item to the set, say $x_{n+1}$, I can say "Let $X = X \cup ...
1
vote
0answers
104 views

How to interpret the sum notation on $\sum_{r:i(r)=i} \langle R_{r\alpha} \rangle^{(t)}$

While doing research for my thesis, I ran into a paper called "Statistical Models for Co-occurrence Data". In the early pages, when talking about an iterative numerical method (a custom EM-method, to ...
9
votes
1answer
742 views

What does the math notation $\sum$ mean?

I have come across this symbol a few times, and I am not sure what it "does" or what it means: $\Large\sum$
0
votes
1answer
130 views

Correct form of sum expression

I want to create equation that represents following piece of code (it is much more complicated I simplify it for clearance) int v = 23; sum = 0; for (int i = v; i > 0; i= i - 2) { sum = sum + i; } ...
3
votes
2answers
62 views

interval for a product to infinity

I was wondering - how would I specify the interval (the amount that n increases each time) between terms? Is that possible? What if I want it to increase by, say, ...