Infinite series Definition and 386 Threads

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...

x+x^2+x^3+x^4... = 14 Find x Could someone please provide an explanation. Thank you

x+x^2+x^3+x^4... = 14 Find x Could someone please provide an explanation on how to solve this?

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...

Evaluate the infinite series $1-\dfrac{2^3}{1!}+\dfrac{3^3}{2!}-\dfrac{4^3}{3!}+\cdots$.

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!

use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. $\sum_{n=1}^{\infty}\frac{1}{3^n+4^n}$

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}...

Expand the quantity (t + P)^(1/2) about 0 in terms of t/P. Give four non-zero terms. (t + P)^(1/2) ~ =

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...

The title pretty much says it all, does anyone know of an infinite series summation that is equal to $$\sqrt{-1}$$?

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...

The title pretty much says it all, does anyone know infinite series summations that are equal to 1/5 or 1/7?

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...

We all know about the sum of the infinite series, 1 + 1/1! + 1/2! + 1/3! + ... to 1/inf! = e What other series do we have that are equal to 'e'?