What is Infinite sums: Definition and 18 Discussions

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.
For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.
In modern terminology, any (ordered) infinite sequence

(

a

1

,

a

2

,

a

3

,

)

{\displaystyle (a_{1},a_{2},a_{3},\ldots )}
of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the ai one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like

or, using the summation sign,

The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series. That is,
When this limit exists, one says that the series is convergent or summable, or that the sequence

(

a

1

,

a

2

,

a

3

,

)

{\displaystyle (a_{1},a_{2},a_{3},\ldots )}
is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent.The notation

i
=
1

a

i

{\textstyle \sum _{i=1}^{\infty }a_{i}}
denotes both the series—that is the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generalization of the similar convention of denoting by

a
+
b

{\displaystyle a+b}
both the addition—the process of adding—and its result—the sum of a and b.
Generally, the terms of a series come from a ring, often the field

R

{\displaystyle \mathbb {R} }
of the real numbers or the field

C

{\displaystyle \mathbb {C} }
of the complex numbers. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product.

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1. A Is the following sum a part of any known generalized function?

I have a sum that looks like the following: ## \sum_{k = 0}^{\infty} \left( \frac{A}{A + k} \right)^{\eta} \frac{z^k}{k!} ## Here, A is positive real. If \eta is an integer, this can be written as: ## \sum_{k = 0}^{\infty} \left( \frac{A(A +1)(A+2) \cdots (A + k - 1)}{(A + 1)(A+2)(A+3)...
2. Looking for a particular function

TL;DR Summary: I want to find a function with f'>0, f''<0 and takes the values 2, 2^2, 2^3, 2^4,..., 2^n Hello everyone. A professor explained the St. Petersburgh paradox in class and the concept of utility function U used to explain why someone won't play a betting game with an infinite...
3. I Discrete mathematics--An easy doubt on the notations of sums

I have a doubt about the notation and alternative ways to represent the terms involved in sums. Suppose that we have the following multivariable function, $$f(x,y)=\sum^{m}_{j=0}y^{j}\sum^{j-m}_{i=0}x^{i+j}$$. Now, let ##\psi_{j}(x)=\sum^{j-m}_{i=0}x^{i+j}##. In the light of the foregoing, is...
4. I Struggling with one step to show quantum operator equality

Hello guys, I struggle with one step in a calculation to show a quantum operator equality .It would be nice to get some help from you.The problematic step is red marked.I make a photo of my whiteboard activities.The main problem is the step where two infinite sums pops although I work...
5. I The product of 2 infinite sums

Hi. I know that eixe-ix = 1 but if I write the product of the 2 exponentials as infinite series I get ΣnΣm xn/(n!) (-x)m/(m!) without knowing the result is 1 using exponentials how would I get the result of this product of 2 infinite sums ? Thanks
6. What Is the Value of the Infinite Sum $$3 \sum_{k=1}^\infty \frac{1}{2k^2-k}$$?

The problem I'd like to calculate the value of this sum: $$3 \sum^\infty_{k=1}\frac{1}{2k^2-k}$$The attempt ## 3 \sum^\infty_{k=1}\frac{1}{2k^2-k} = [k=t/2] = 3 \sum^\infty_{t=2}\frac{1}{2 \left( \frac{t}{2} \right)^2-\frac{t}{2}} = 3 \sum^\infty_{t=2}\frac{1}{ \frac{t^2}{2} - \frac{t}{2}} = 3...
7. General solution for the heat equation of a 1-D circle

Homework Statement Modify the initial conditions (for the diffusion equation of a circle) to have the initial conditions ## g(\theta)= \sum_{n=-\infty}^{\infty}d_{n}e^{2\pi in\theta} ## Using the method of Green's functions, and ## S(\theta,t)= \frac{1}{\sqrt{4\pi...
8. I Product of complex conjugate functions with infinite sums

Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) ,$$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and...

12. Does the sum of ln(k/(k+1)) converge or diverge as n approaches infinity?

So I was trying to see if \Sigmaln(\frac{n}{n+1}) diverges or converges. To see this I started writing out [ln(1) - ln(2)] + [ln(2) - ln(3)] + [ln(4) - ln(5)] ... I noticed that after ln(1) everything must cancel out so I reasoned that the series must converge on ln(1) which equals ZERO...
13. Is it possible to transform infinite sums into infinite products?

is it also possible to transform any these kinds summation to any product notation: 1. infinite - convergent 2. infinite - divergent 3. finite (but preserves the "description" of the sequence) For example, I could describe the number 6, from the summation of i from i=0 until 3. Could I...
14. M

Swapping Integrals and Sums: When is it Justifiable?

when using the reimann integral over infinite sums, when is it justifiable to interchange the integral and the sum? \int\displaystyle\sum_{i=1}^{\infty} f_i(x)dx=\displaystyle\sum_{i=1}^{\infty} \int f_i(x)dx thanks ahead for the help!
15. Help With Partial Derivatives and Infinite Sums

I'm working on a calculus project and I can't seem to work through this next part... I need to substitute equation (2) into equation (1): (1): r\frac{\partial}{\partial r}(r\frac{\partial T}{\partial r})+\frac{\partial ^{2}T}{\partial\Theta^{2}}=0 (2): \frac{T-T_{0}}{T_{0}}=A_{0}+\sum from n=1...
16. Trying to find the quotient of infinite sums

i am trying to re-express the following in terms of a rational function: \frac{(0+x+2x^2+3x^3+...)}{1+x+x^2+x^3+...} . i know that this is supposed to be \frac{1}{x-1} but I can't figure out how to do it. I know the denominator is just \frac{1}{1-x}. so in order for this work out, the...
17. Limits of infinite sums of sequences

I understand that the limit of the sum of two sequences equals the sum of the sequences' limis: \displaysyle \lim_{n\rightarrow\infty} (a_{n} + b_{n}) = \lim_{n\rightarrow\infty}a_{n} + \lim_{n\rightarrow\infty}b_{n}. Similar results consequenly hold for sums of three sequences, four sequences...
18. Integrating infinite sums and macluarin's expansion

Homework Statement Using the macluarin's expansion for sinx show that \int sinx dx=-cosx+cHomework Equations sinx=\sum_{n=0} ^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!} The Attempt at a Solution Well I can easily write out some of the series and just show that it is equal to -cosx but if I...