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.
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...
Homework Statement
Hi, I have to find the RMS value of the inifnite series in the image below.
Homework Equations
https://en.wikipedia.org/wiki/Cauchy_product
Allowed to assume that the time average of sin^2(wt) and cos^2(wt) = 1/2
The Attempt at a Solution
So to get the RMS value I think I...
The problem is to find the general term ##a_n## (not the partial sum) of the infinite series with a starting point n=1
$$a_n = \frac {8} {1^2 + 1} + \frac {1} {2^2 + 1} + \frac {8} {3^2 + 1} + \frac {1} {4^2 + 1} + \text {...}$$
The denominator is easy, just ##n^2 + 1## but I can't think of...
Homework Statement
Show that ##\sum_{n=1}^{\infty}\frac{1}{n^{4}}=\frac{\pi^{4}}{90}##.
Homework Equations
The Attempt at a Solution
##\frac{1}{n^{4}} = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots##.
Do I now factorise?
Homework Statement
"A dollar due to be paid to you at the end of n months, with the same interest rate as in Problem 13, is worth only (1.005)^{-n} dollars now (because that is what will amount to $1 after n months). How much must you deposit now in order to be able to withdraw $10 a month...
I learn that we can expand the electric potential in an infinite series of rho and cos(n*phi) when solving the Laplace equation in polar coordinates. The problem I want to consider is the expansion for the potential due to a 2D line dipole (two infinitely-long line charge separated by a small...
In my book, applied analysis by john hunter it gives me a strange way of stating an infinite sum that I'm still trying to understand because in my calculus books it was never described this way.
It says:
We can use the definition of the convergence of a sequence to define the sum of an...
The text does it thusly:
imgur link: http://i.imgur.com/Xj2z1Cr.jpg
But, before I got to here, I attempted it in a different way and want to know if it is still valid.
Check that f^{*}f is finite, by checking that it converges.
f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x +...
Hello,
I have been reviewing my textbook lately, and I came across a rather paradoxical statements. all of the convergence tests in my book state that the terms of the series has to be positive. However, when I solved this power series ∑(-1)n-1(xn/n), I found that it converges for -1<x≤1, but...
So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered
e^{At} \vec{u}(0) = \vec{u}(t)
as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...
Homework Statement
hello
i have a question to this solved problem in the book
" Mathematical Methods for Physics and Engineering Third Edition K. F. RILEY, M. P. HOBSON and S.J. BENCE "
page 118
Consider a ball that drops from a height of 27 m and on each bounce retains only a third
of its...
I am currently self studying the book and I think it is a great book and it really does go deep into the subject. But about how to study this book i am trying to prove every theorem that i come across and it is new. To get the most out of this book i would like to know other opinions on how to...
Homework Statement
[/B]
Hello, this problem is from a well-known calc text:
Σ(n=1 to ∞) 8/(n(n+2)Homework Equations
[/B]
What I have here is decomposingg the problem into Σ(n=1 to ∞)(8/n -(8/n+2)The Attempt at a Solution
I have the series sum as equaling (8/1-8/3) + (8/2-8/4) + (8/3-8/5) +...
Hey everyone,
I'm currently in Calc 2 and the only thing I seem to be having a problem with is a couple of the convergence tests. When I take pretty much any math course, I always use mathematica to help check my answers when I'm doing HW or practicing so I don't waste time. My question is...
Homework Statement
Determine whether the series is converging or diverging
Homework Equations
∞
∑ 1 / (3n +cos2(n))
n=1The Attempt at a Solution
I used The Comparison Test, I'm just not sure I'm right. Here's what I've got:
The dominant term in the denominator is is 3n and
cos2(n)...
Homework Statement
I came across a problem where f: (-π/2, π/2)→ℝ where f(x) = \sum\limits_{n=1}^\infty\frac{(sin(x))^n}{\sqrt(n)}
The problem had three parts.
The first was to prove the series was convergent ∀ x ∈ (-π/2, π/2)
The second was to prove that the function f(x) was continuous...
Homework Statement
Part a.) For a>0 Determine Limn→∞(a1/n-1)
Part b.) Now assume a>1
Establish that Σn=1∞(a1/n-1) converges if and only if Σn=1∞(e1/n-1) converges.
Part c.) Determine by means of the integral test whether Σn=1∞(e1/n-1) converges
Homework Equations
Integral Test
Limit...
Homework Statement
For the following series ∑∞an determine if they are convergent or divergent. If convergent find the sum.
(ii) ∑∞n=0 cos(θ)2n+sin(θ)2n[/B]Homework Equations
geometric series, [/B]The Attempt at a Solution
First I have to show that the equation is convergent.
Both cos(θ)...
Not all DEs have a closed form solution. Some DEs have an implicit solution only - you cannot algebraically solve one variable of interest for another.
I have seen on this forum people solving DEs in terms of infinite series. How does one arrive at such a solution, and can an implicit...
Sum= ...- 1 + 1 -1 +1-1+1... until infinite
It is just an infinite sum of -1 plus 1.
Can anyone tell me the sum of this infinite series and a demonstration of that result?
THanks!
Homework Statement
Find the expectation value of the Energy the Old Fashioned way from example 2.2.
Homework Equations
##\left< E \right> =\frac { 480\hbar ^{ 2 } }{ \pi ^{ 4 }ma^{ 2 } } \sum _{ odds }^{ \infty }{ \frac { 1 }{ { n }^{ 4 } } } ##
The Attempt at a Solution
Never...
Homework Statement
Recognize the series $$3-3^3/3!+3^5/5!-3^7/7!$$ is a taylor series evaluated at a particular value of x. Find the sumHomework Equations
Sum of Infinite series = ##a/1-x##
The Attempt at a Solution
So, I can't figure out what i would us as the ratio (the thing you multiply...
Hello,
I've been reviewing some calculus material lately and I just have a couple questions:
1) I've seen infinite series shown graphically as a collection of rectangular elements under a curve representing an approximation of the area under the curve. But the outputs of the infinite...
In certain forms - including the logarithmic - a number of the trigonometric and hyperbolic functions can be used to sum series having Riemann Zeta and Dirichlet Beta functions (in the general series term). In this tutorial, we explore some of these connections, and present a variety of Zeta and...
Homework Statement
Prove that:
[n+1 / n^2 + (n+1)^2 / n^3 + ... + (n+1)^n / n ^ (n+1) -> e-1
Homework Equations
I have been trying it for couple of days. Tried to work the terms, natural log it all, use the byniomial theory but i can´t get to the right answer.
The Attempt at a...
Hello!
How can I justify that the infinite series 1 - 1 + 1 - 1 + 1 - 1... is divergent?
If I were to look at this, I see every two terms canceling out and thus, and assume that it is convergent since the sum doesn't blow up. That's what my intuition would tell me.
I know I can use...
Homework Statement
Find the value of An and given that f(x) = 1 for 0 < x < L/2, find the sum of the infinite series.
Homework Equations
The Attempt at a Solution
The basis is chosen to be ##c_n = \sqrt{\frac{2}{L}}cos (\frac{n\pi }{L}x)## for cosine, and ##s_n = \sqrt{\frac{2}{L}}sin...
Problem:
If $0<x<1$ and
$$A_n=\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+\cdots +\frac{x^{2^n}}{1-x^{2^{n+1}}}$$
then find $\displaystyle \lim_{n\rightarrow \infty}A_n$.
Attempt:
I tried to see if it can be converted to a telescoping series but I had no luck. Then, I tried this:
$$\lim_{n\rightarrow...
Homework Statement
I just have to graph this function to see where the "Gibbs phenomenon" occurs in its Fourier Series representation. I am pretty sure I integrated correctly.Homework Equations
Fourier Series
The Attempt at a Solution...
Homework Statement
This is for Calculus II. We've just started the chapter on Infinite Series. n runs from 1 to ∞.
\Sigma\frac{1}{n(n+3)}
The Attempt at a Solution
I used partial fraction decomposition to rewrite the sum.
\frac{1}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}...
I'm not sure which category this post actually belongs to, or if the title of this post is even accurate. I guessed Calculus was the closest one.
I watched this video on the web after a professor told me this mathematical phenomenon (http://www.youtube.com/watch?v=w-I6XTVZXww). It asserts...
Homework Statement
I have been working on a truncated Fourier series. I have come up with a truncated series for cos (αx) and it matches my book, where in this case I'm letting x=π, and then I have shown as the book asks,
Truncated series = F_{N}(π) = cos (απ) + \frac{2α}{π}...
This might seem like a rudimentary question but when trying to prove divergence (or even convergence) of an infinite series does the series always have to start at n = 1?
For example would doing a test for \sum^{∞}_{n=1}\frac{1}{n} be any different from \sum^{∞}_{n=0}\frac{1}{n}
Question:
I was just wondering if there was any error in what I've done in the following steps to find the series representation of ##lnx##. I know ## \frac {1}{x}## is given in the following link by doing having the a function centred at 0, you can let ##f(x) = ∑^∞_{n=0} \frac...
Homework Statement
Problem # 30 in Ch1 Section 16 in Mary L. Boas' Math Methods in the Physical Sciences
It is clear that you (or your computer) can’t find the sum of an infinite series
just by adding up the terms one by one. For example, to get \zeta (1.1)=\sum _{ n=1 }^{ \infty }{...
I have a hard time believing we only have the limited number of series I have seen so far especially considering how much broader mathematics is than I had thought just a short while ago.
Where should I search to find more infinite series summations for the gamma function? For example which...
Does anyone know if we currently have an infinite series summation general solution for the gamma function such as,
$$\frac{1}{\Gamma(k)}=\sum_{n=0}^{\infty} f(n,k)$$
or,
$${\Gamma(k)}=\sum_{n=0}^{\infty} f(n,k)$$?
Homework Statement
Use Maclaurin’s theorem to derive the first five terms of the series expansion for ##(1+x)^{r}##, where -1<x<1. Assuming the series, obtained above, continues with the same pattern, sum the following infinite series
##1 + \frac{1}{6} - \frac{(1)(2)}{(6)(12)} +...
So the title pretty much says it all, what other infinite series summations do we have for Pi^2/6 besides,
$$\sum_{n=1}^{\infty} 1/n^2$$
***EDIT*** I should also include,
$$\sum_{n=1}^{\infty} 2(-1)^(n+1)/n^2$$
$$\sum_{n=1}^{\infty} 4/(2n)^2$$
etc. etc.
A unique form outside of the 1/n^2 family.
I have a series that takes steps of '2' which requires an operation starting from n=1 to do the following,
@n=1 (n-1)!
@n=3 (n-3)!-(n-1)!
@n=5 (n-5)!-(n-3)!+(n-1)!
@n=7 (n-7)!-(n-5)!+(n-3)!-(n-1)!
etc. etc.
Any ideas?
*EDIT*
Come to think of it,
This problem would probably be easier to...
Homework Statement
let bk>0 be real numbers such that Ʃ bk diverges. Show that the series Ʃ bk/(1+bk) diverges as well.
both series start at k=1Homework Equations
From the Given statements, we know 1+bk>1 and 0<bk/(1+bk)<1
The Attempt at a Solution
I've tried using comparison test but cannot...
Given,
U(n)=1/(logn)^(2*n)
To find:
Whether the series ƩU(n) is convergent or divergent.
Sequence of tests to be followed:
*Comparison tests
*Integral tests
*D'Alembert's ratio test
*Raabe's test
*Logarithmic test
*Cauchy's root test
My approach:
Comparison test:
Since the series V(n) cannot...