Series Definition and 998 Threads

i wanted to ask you that i am having financial problem for preparing for exams for getting admission into international universities. are there online test series for physics,chemistry,maths in free

<<Moderator's note: moved from a technical forum, so homework template missing.>> I found a problem in Boas 3rd ed that asks the reader to use S_n = 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + ... to show that the harmonic series diverges. They specifically want this done using the test...

I'm not sure this is the right forum, so if not, please move to the appropriate forum. My question is why does the ratio of two consecutive fibonacci numbers converge to the golden ratio? I see no mathematical connection between the series ratios and ratios of a unit line segment divided into...

I am playing with Arduino and LEDs at the moment. LED needs a resistor to limit current, that's clear. However, all examples I see use separate resistor for each diode. As far as I can tell electrically (in terms of limiting current) it shouldn't matter much whether we use single resistor for...

Homework Statement I Have a differential equation y'' -xy'-y=0 and I must solve it by means of a power series and find the general term. I actually solved the most of it but I have problem to decide it in term of a ∑ notation! Homework Equations y'' -xy'-y=0 The Attempt at a Solution I know...

I'm wondering if anyone could give me the intuition behind Fourier series. In class we have approximated functions over the interval ##[-\pi,\pi]## using either ##1, sin(nx), cos(nx)## or ##e^{inx}##. An example of an even function approximated could be: ## f(x) = \frac {(1,f(x))}{||1||^{2}}*1...

I have been having trouble getting the calculation of energy for a chain of coupled oscillators to come out correctly. The program was run in Matlab and is intended to calculate the energy of a system of connected Hooke's law oscillators. Right now there is only stiffness and no dampening...

I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how. $$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$ I get most of the function, I just can't see where the ##-1## comes from. Could...

From Mary Boas' "Mathematical Methods in the Physical Sciences" Third Edition. I'm not taking this class but I was going through the textbook and ran into an issue. The problem states: If you invest a dollar at "6% interest compounded monthly," it amounts to (1.005)n dollars after n months. If...

Homework Statement Find the power series in x-x0 for the general solution of y"-y=0; x0=3. Homework Equations None. The Attempt at a Solution I'll post my work by uploading it.

Homework Statement Perform a Taylor Series expansion for γ in powers of β^2, keeping only the third terms (ie. powers up to β^4). We are assuming at β < 1. Homework Equations γ = (1-β^2)^(-1/2) The Attempt at a Solution I have no background in math so I do not know how to do Taylor expansion...

Homework Statement An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants. a) Find v(t) and x(t). b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3. c) Find the object’s terminal velocity. Homework...

Homework Statement Prove that, $$\sum _{n=1,3,5...} \frac{1}{n} e^{-nx} \sin{ny} = \frac{1}{2}\tan^{-1} (\frac{\sin{y}}{\sinh{x}})$$ Homework Equations $$\tan^{-1}{x} = x - \frac{x^3}{3} +\frac{x^5}{5} - ... $$ 3. The Attempt at a Solution $$\sum _{n=1,3,5...} \frac{1}{n} e^{-nx}...

Homework Statement Note - I do not know why there is a .5 after the ampere. I think it is an error and I have asked my lecturer to clarify. Homework Equations The Attempt at a Solution f(t)=sint2 f(0)=sin(0)2=0 f'(t)=2sintcost f'(0)=sin2(0)=0...

Homework Statement The following series are not power series, but you can transform each one into a power series by a change of variable and so find out where it converges. ∑∞0 ((3n(n+1)) / (x+1)n Homework Equations a power series is a series of the form: a0 + a1x + a2x^2 ... + ... The...

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...

Homework Statement "For the given series, write formulas for the sequences an , Sn, Rn and find the limit as n->∞ (if it exists) Homework Equations ∑∞1 ((1/n) - 1/(n+1) The Attempt at a Solution I know how to take the limit, that's no problem. I'm a bit confused about what an , Sn, Rn are...

I'm not sure if this should go in the homework forum or not, but here we go. Hello all, I've been trying to find a series representation for the elliptic integral of the first kind. From some "research", the power series for the complete form (## \varphi=\frac{\pi}{2} ## or ## x=1 ##) seems to...

Homework Statement Homework Equations Summation The Attempt at a Solution I know I could have simplified (3n-2)^3 +(3n-1)^3 -(3n)^3 and put the formulas in but I wonder is there any other method (I was thinking about grouping the terms, but to no avail) to work this out.

This is a desperate attempt to find a set of videos I saw about a year ago on YouTube. It was not one of the big, well known guys like Veritasium or SciShow, it was just one middle aged guy. He explained scientific advancements through history, and gave really, really detailed accounts of how...

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...

Hi guys, I am doing this question of alternating series test. And I was following the below principles when solving the problem. Sorry I don't know how to type in the math language. I got 4, 8, 9, 10 as the answers. But the system rejected this without any explanation. Can someone throw a...

Calculate the sum for the infinite geometric series $4+2+1+\frac{1}{2}+...$ all I know is the ratio is $\frac{1}{2}$ $\displaystyle\sum_{n}^{\infty}a{r}^{n}$ assume this is used

Homework Statement i have a few homework question and want to be sure if I have solved them right. Q1) Write ##\vec{\triangledown}\cdot\vec{\triangledown}\times\vec{A}## and ##\vec{\triangledown}\times\vec{\triangledown}\phi## in tensor index notation in ##R^3## Q2) the spherical coordinates...

Homework Statement I was given a problem with a list of sums of sinusoidal signals, such as Example that I made up: x(t)=cos(t)+5sin(5*t). The problem asks if a given expression could be a Fourier expansion. Homework Equations [/B]The Attempt at a Solution My guess is that it has something to...

The books are based on Schwinger's but is much easier read. Uses my favorite spins-first approach. Lectures On Quantum Mechanics vol. 1, 2, & 3 by Berthold-Georg Englert https://www.amazon.com/dp/9812569715/?tag=pfamazon01-20 https://www.amazon.com/dp/9812569731/?tag=pfamazon01-20...

Hello, i am not sure where to discuss it but here maybe proper for this thread. I just want to discuss about DC's Tv show Flash and physics on it like singularity or parallel universes?

Homework Statement Homework Equations no equations required 3. The Attempt at a Solution a) so for part c) i came up with two formula's for the tortoise series: the first formula (for the toroise series) is Sn = 20n This formula makes sense and agrees with part a). for example, if the...

So say we had 2 batteries, B1 and B2, and B2 is on top of B1. The + terminal of B1 connects to - terminal of B2, and the + terminal of B2 connects to - terminal of B1. Why does this double the voltage compared to the voltage of just B1?

So here is the problem I am trying to solve: You can combine two (or more) convergent power series on the same interval I. Using the properties of the geometric series, find the power series of the function below. Series: f(x) = 1/(1 - x) = sigma k = 0, infinity = 1+ x + x^2 + x^3 Function...

I need to find the Maclaurin series for $$f(x) = x^2e^x$$ I know $$e^x = \sum_{n = 0}^{\infty} \frac{x^n}{n!}$$ So, why can't I do $$x^2 e^x =x^2 \sum_{n = 0}^{\infty} \frac{x^n}{n!} = \sum_{n = 0}^{\infty} \frac{x^2 x^n}{n!} $$

I am linearizing a vector equation using the first order taylor series expansion. I would like to linearize the equation with respect to both the magnitude of the vector and the direction of the vector. Does that mean I will have to treat it as a Taylor expansion about two variables...

Homework Statement Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0. Homework Equations None. The Attempt at a Solution I'll post my whole work.

I discovered this interesting series of videos that others might appreciate. Includes interviews with Alan Guth, Roger Penrose and loads of other interesting people. Definitely not pop-sci and fairly up to date. Episode 1 is a bit low quality with unnecessary subtitles, but it gets better as it...

Homework Statement Find a power series that represents $$ \frac{x}{(1+4x)^2}$$ Homework Equations $$ \sum c_n (x-a)^n $$ The Attempt at a Solution $$ \frac{x}{(1+4x)^2} = x* \frac{1}{(1+4x)^2} $$ since \frac{1}{1+4x}=\frac{d}{dx}\frac{1}{(1+4x)^2} $$ x*\frac{d}{dx}\frac{1}{(1+4x)^2}...

Homework Statement I'm calculating the coefficients for the Fourier series and I got to part where I can't simplify an any further but I know I have to. a_n = \frac{1}{2π}\Big[\frac{cos(n-1)π}{n-1}-\frac{cos(n+1)π}{n+1}-\frac{1}{n-1}+\frac{1}{n+1}\Big]Homework EquationsThe Attempt at a...

I'm really confused about this test. Suppose we let f(n)=an and f(x) follows all the conditions. When you take the integral of f(x) and gives you some value. What are you supposed to conclude from this value?

Homework Statement By applying the Gram–Schmidt procedure to the list of monomials 1, x, x2, ..., show that the first three elements of an orthonormal basis for the space L2 (−∞, ∞) with weight function ##w(x) = \frac{1}{\sqrt{\pi}} e^{-x^2} ## are ##e_0(x)=1## , ##e_1(x)= 2x## ,##e_2(x)=...

I need to use the maclaurin series to find where this series converges: $$\sum_{n = 0}^{\infty} (-1)^n \frac{\pi^{2n}}{(2n)!}$$ But I'm not sure how to do this.

I need to find the function for this Maclaurin series $$1 - \frac{5^3x^3}{3!} + \frac{5^5x^5}{5!} - \frac{5^7x^7}{7!} ...$$ I can derive this sigma: $$1 + \sum_{n = 2}^{\infty} \frac{(-1)^{n - 1} 5^{2n - 1} x^{2n - 1}}{(2n - 1)!}$$ But I'm not sure how to get this function from this series.

I need to find the maclaurin series of the function $$\frac{1}{1 - 2x}$$. I know $\frac{1}{1 - x}$ is $1 + x + x^2 + x^3 ...$ but how can I use this to solve the problem? I don't think I can just plug in $2x$ can I?

Aren't the Maclaurin series an expansion of a function about 0 f(x) = f(0) + (f '(0) / 1!) * x + (f ''(0) / 2!) * x^2 + (f '''(0) / 3!) * x^3 + ...

I need to find the Maclaurin series for $$f(x) = e^{x - 2}$$ I know that the maclaurin series for $f(x) = e^x$ is $$\sum_{n = 0}^{\infty} \frac{x^n}{n!}$$ If I substitute in $x - 2$ for x, I would get $$\sum_{n = 0}^{\infty} \frac{(x - 2)^n}{n!}$$ However, this is wrong, according to the...

I need to find the Maclaurin series of this function: $$f(x) = ln(1 - x^2)$$ I know that $ln(1 + x)$ equals $$\sum_{n = 1}^{\infty}\frac{(-1)^{n - 1} x^n}{n}$$ Or, $x - \frac{x^2}{2} + \frac{x^3}{3} ...$ If I swap in $-x^2$ for x, I get: $$-x^2 + \frac{x^4}{2} - \frac{x^5}{3} +...

I'm examining the Maclaurin series for $f(x) = ln(x + 1)$. It is fairly straightforward but there are a few details I'm not getting. So: $$ ln(x + 1) = \int_{}^{} \frac{1}{1 + x}\,dx$$ which equals: $A + x - \frac{x^2}{2}$ etc. or $A + \sum_{n = 1}^{\infty}(-1)^{n - 1}\frac{x^n}{n}$ I'm...

I need to prove that for $-1 < x < 1$ $$\frac{1}{(1 - x)^2} = 1 + 2x + 3x^2 + 4x^3 ...$$ So, according to the textbook, the geometric series has a radius of convergence $R = 1$ (I'm not sure how this is true). In any case we can compare it to: $$\frac{1}{1 - x} =\sum_{n = 0}^{\infty} x^n$$...

I need to find the Maclaurin series for this function: $$f(x) = (1 - x)^{- \frac{1}{2}}$$ And I need to find $f^n(a)$ First, I need the first few derivatives: $$f'(x) ={- \frac{1}{2}} (1 - x)^{- \frac{3}{2}}$$ $$f''(x) ={ \frac{3}{4}} (1 - x)^{- \frac{5}{2}}$$ $$f'''(x) ={- \frac{15}{8}}...

So I have $$\sum_{n = 2}^{\infty} \frac{1}{nln(n)}$$ I'm trying to apply the limit comparison test, so I can compare it to $b_n$ or $\frac{1}{n}$ and I can let $a_n = \frac{1}{nln(n)}$ Then I get $$\lim_{{n}\to{\infty}} \frac{n}{nln(n)}$$ Or $$\lim_{{n}\to{\infty}} \frac{1}{ln(n)}$$ Which is...

I have this series $$\sum_{n =0}^{\infty}\frac{(-1)^n {x}^{2n}}{{2}^{n + 1}}$$ I need to find whether it converges or diverges at $\sqrt{2}$ and $-\sqrt{2}$. I'm not quite sure how to approach this. For $\sqrt{2}$ I have $$\sum_{n =0}^{\infty}\frac{(-1)^n {\sqrt{2}}^{2n}}{{2}^{n +...