What is Infinite series: Definition and 389 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.
Consider the geometric series ##\sum\limits_{n=1}^\infty r^n##.
This infinite series comes from the sequence ##\{g_n\}=\{1,r,r^2,r^3,\ldots\}##.
If ##|r|\geq 1## then ##\lim\limits_{n\to\infty} g_n=\infty\neq 0##. This limit shows that the geometric series does not converge. That is...
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)...
##\sum_{n=1}^\infty n^{-a}## converge s for ##a\gt 1## - otherwise diverges. Is there any theory for ##a_n##? For example ##a_n\gt 1## and ##\lim_{n\to \infty} a_n =1##. How about non-convergent with ##\liminf a_n=1##?
##s_1=2##
##s_2=4##
##s_3=5.333##
##s_4=5.9999##
##(s_n)## is increasing, but unable to guess a bound. Let's see if Cauchy criterion can do something.
For n>2,
$$
s_{n+k} - s_n = \frac{2^{n+1} }{(n+1)!} + \frac{ 2^{n+2} }{(n+2)!} + \cdots \frac{2^{n+k} }{(n+k)!}
$$
$$
s_{n+k} - s_n <...
How to prove that \[ \sum_{i=1}^{\infty}\frac{1}{2^{3i}}\left(\csc^{2}\left(\frac{\pi x}{2^{i}}\right)+1\right)\sec^{2}\left(\frac{\pi x}{2^{i}}\right)\sin^{2}\left(\pi x\right)=1 \] for all \( x\in\mathbb{R} \).
Using graph, we can see that the value of this series is 1 for all values of x...
Summary:: How to know which one is bigger when n goes to infinity?
$$ \sum_{n=1}^\infty \frac {1} {\sqrt {n}(\sqrt {n+1}+\sqrt {n-1})} $$
And:
$$ \sum_{n=1}^\infty \frac {1} {\sqrt {n}(\sqrt {n}+\sqrt {n})} $$
I thought at first that the second one is bigger, although, I came to realize, to my...
Mary Boas attempts to explain this by pointing out that the situation cannot arise because charges will have to be placed individually, and in an order, and that order would represent the order we sum in. That at any point the unplaced infinite charges would form an infinite divergent series...
I'm trying to get from the formula in the top to the formula in the bottom (See image: Series). My approach was to complexify the sine term and then use the fact that (see image: Series 1) for the infinite sum of 1/ne^-n. Then use the identity (see image: Series 2). Any other ideas?
While transforming the equation of the Basel problem, the following infinite series was obtained.
$$\sum_{n=1}^{\infty} \frac{n^2+3n+1}{n^4+2n^3+n^2}=2$$
However I couldn't think of a simple way to prove that.
Can anyone prove that this equation holds true?
I found that ρn = √(2n+1)/(n+1).
Then, I found ρ = lim when n→∞ |(1/n) (√(2n+1))/((1/n) (n+1))| = 0
Based on this result I concluded the series converges; however, the book answer says it diverges. What am I doing wrong?
After evaluating the integral I found the following:
(1/3)tan-1(e∞/3) = (1/3)tan-1(∞) = (1/3)(nπ/2), where n is an odd number. In this case I found multiple solutions to the problem. How do you prove it converges?
I got the following expression:
-(1/4)ln((n+2)/(n-2))
When I substitute "∞" in the expression I found it undefined. However, the book says the series converges. What am I doing wrong?
Hi,
I have a particular equation in a paper, wherein the author specifies an infinite series. The author has apparently found the sum of the series and calculated the equation. Can anyone please help me in understanding how to sum such a series. I have attached part of the paper with the...
Hi, I'm trying to solve the sum of following infinite series:
\sum_{k=1}^{\infty} \frac{{k}^{2}+4}{{2}^{k}} = \sum_{k=1}^{\infty} \frac{{k}^{2}}{{2}^{k}}
+ \sum_{k=1}^{\infty} \frac{4}{{2}^{k}}
Using partial sum we can rewrite the first series: \sum_{k=1}^{\infty}...
To anyone that can help me with this - You have to pick the FIRST correct reason. Work below (exception of 4 because I cannot figure it out), but in order to get the question right you must have all correct and I cannot figure it out. Any help is appreciated.
[Moderator's note: Moved from a...
Evaluate ##\lim_{n \rightarrow +\infty} \frac {1} {n} [(\frac {1}{n})^{1.5} + (\frac {2}{n})^{1.5} +(\frac {3}{n})^{1.5}+ (\frac {4}{n})^{1.5}+...+(\frac {n}{n})^{1.5}]##
Hello. So I'm solving this question at the moment. I know I'm supposed to find out the summation of this before being able...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help In order to formulate a rigorous proof to the proposition stated in Exercise 2.3.10 (1) ... ...
Exercise 2.3.10 (1) reads as...
The problem of the interaction of a point charge with a dielectric plate of finite thickness implies the existence of an infinite series of image charges (see http://www.lorentzcenter.nl/lc/web/2011/466/problems/2/Sometani00.pdf). I introduce notations identical to those used in this work. The...
For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
Homework Statement
Find the sum of the series
Homework EquationsThe Attempt at a Solution
Not sure exactly where to start. If I move 3 outside the sum I'm left with 3*sigma(1/n*4^n), which I can rewrite to 3*sigma((1/n)*(1/4)^n), which party looks like a geometric series..Any tips?
For a series to be convergent,it must have a finite sum,i.e.,limiting value of sum.As the sum of n terms approaches a limit,it means that the nth term is getting smaller and tending to 0,but why is not the converse true?Should not the sum approach a finite value if the nth term of the series is...
Is there a simple closed-form solution for the following infinite series?
##F(a,b,c) = \sum_{j=0}^\infty \frac{(j+a)!}{(j+b)! (j+c)!}##
where ##a, b, c## are positive integers?
While I was was numerically integrating the magnetic field caused by an infinite array of magnetic moments, I observed the interesting limit ( limit (1) in the image). It may seem difficult to prove it mathematically but from the physics point of view, I think it can be proved relatively...
Homework Statement
Homework EquationsThe Attempt at a Solution
So the book is saying that this series diverges, i have learned my lesson and have stopped doubting the authors of this book but i don't understand how this series diverges. ok i can use the comparison test using 1/3n and 1/3n...
Homework Statement
Homework EquationsThe Attempt at a Solution
So my understanding of this so far is that the whole infinite series from 1 to infinities summation minus the first six terms summation is equal to 0.0002..? This is so confusing. So how does that mean that the sum will lie...
I'm trying to make an approximation to a series I'm generating; the series is constructed as follows:
Term 1:
\left[\frac{cos(x/2)}{cos(y/2)}\right]
Term 2:
\left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right]
I'm not sure yet if the series repeats itself or forms a pattern...
Homework Statement
Trying to find the sum of (-1)3n+1/(2n-1)3. by using term-by-term integration on the cosine Fourier series x= L/2-4L/π2∑cos(((2n-1)πx)/L)/(2n-1)2.
Homework Equations
Shown below
The Attempt at a Solution
When integrating and substituting Lx/2 for x's sine Fourier series I...
This is my first time posting so forgive me if I have it in the wrong place,
i'm trying to find a solution to the following that I can stick into either excel or a VBA script. It has been 25 years since I looked at any serious maths and I'm stumped. I can find and digest e^-(n^2y) but can't...
Homework Statement
and in this case we have,
[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries_files/eq0016MP.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries_files/empty.gif
Homework Equations
I can not see how they get either of...
Hey guys! I just have a question regarding finding the sum of an infinite series. Attached is the image of the question. I've tried to use the ratio test but it doesn't give me the result I need which happens to be 1/sqrt(2). I feel like this is one of those power series questions, but I'm not...
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...
Homework Statement
Find the average energy ##\langle E \rangle## for
(a) an n-state system in which a given state can have energy 0, ε, 2ε, 3ε... nε.
(b) a harmonic oscillator, in which a state can have energy 0, ε, 2ε, 3ε... (i.e. with no upper limit).
Homework Equations
Definition of...
Homework Statement
Hello,
I am currently doing some holiday pre-study for signals analysis coming up next semester. I'm mainly using MIT OCW 6.003 from 2011 with some other web resources (youtube, etc).
The initial stuff is heavy on the old infinite series stuff, that seems often skimmed...
[Please excuse the screengrabs of the fomulae - I'll get around to learning TeX someday!]
1. Homework Statement
Find the sum of this series (answer included - not the one I'm getting)
The Attempt at a Solution
So I'm trying to sum this series as a telescoping sum. I decomposed the fraction...
Homework Statement
A fishery manager knows that her fish population naturally increases at a rate of 1.4% per month, while 119
fish are harvested each month. Let Fn be the fish population after the nth month, where F0 = 4500 fish. Assume that that process continues indefinitely. Use the...
It seems to me that convergence rounds away the possibility of there being a smallest constituent part of reality.
For instance, adding 1/2 + 1/4 + 1/8 . . . etc. would never become 1, since there would always be an infinitely small fraction that made the second half unreachable relative to the...
Homework Statement
Find the sum of the given infinite series.
$$S = {1\over 1\times 3} + {2\over 1\times 3\times 5}+{3\over 1\times 3\times 5\times 7} \cdots $$
2. Homework Equations The Attempt at a Solution
I try to reduce the denominator to closed form by converting it to a factorial...
Question
∞
∑ tan(1/n)
n = 1
Does the infinite series diverge or converge?
Equations
If limn → ∞ ≠ 0 then the series is divergent
Attempt
I tried using the limit test with sin(1/n)/cos(1/n) as n approaches infinity which I solved as sin(0)/cos(0) = 0/1 = 0
This does not rule out anything and I...
I have a set of data that I've been working with that seems to be defined by the sum of a set of exponential functions of the form (1-e^{\frac{-t}{\tau}}). I've come up with the following series which is the product of a decay function and an exponential with an increasing time constant. If this...
Hi folks,
I was wondering if there are books that explain how to solve differential equations using infinite series. I know it is possible to do it since Poincaré used that method.
Do you know which ones are the best?
I find books on infinite series but they talk just about series...
Homework Statement
To Determine Whether the series seen below is convergent or divergent.
Homework Equations
∑(n/((n+1)(n+2))) From n=1 to infinity.
The Attempt at a Solution
Tried to use the comparison test as the bottom is n^2 + 3n + 2, comparing to 1/n. However, this does not work as the...
Hello - I'm not sure this is where this should go, but I'm working with Laplace Transforms and differential equations, so this seems as good a place as any. Also, I doubt this is graduate level math strictly speaking, but I went about as high as you can go in calculus and linear algebra during...
Homework Statement
The problem states: In the harmonic series ##\sum_{1}^{\infty} \frac{1}{k}##, all terms for which the integer ##k## contains the digit 9 are deleted. Show that the resulting series is convergent.
Hint: Show that the number of terms ##\frac{1}{k}## for which ##k## contains no...
a. Find the common ration $r$, for an infinite series
with an initial term $4$ that converges to a sum of $\displaystyle\frac{16}{3}$
$$\displaystyle S=\frac{a}{1-r} $$ so $\displaystyle\frac{16}{3}=\frac{4}{1-r}$ then $\displaystyle r=\frac{1}{4}$
b. Consider the infinite geometric series...