Series Definition and 998 Threads

  1. Imaxx

    B How to prove this infinite series?

    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?
  2. Imaxx

    Is This Infinite Series Convergent?

    ∞ ∑ (n∧2+3n+1) / (n∧4+2n∧3+n∧2) =? n=1 I attempted to find the general sum of this 'expression'?? But no luck. How can I solve this?
  3. T

    MHB How to estimate simplex gradient using Taylor series?

    I read Iterative methods for optimization by C. Kelley (PDF) and I'm struggling to understand proof of Notes on notation: S is a simplex with vertices x_1 to x_{N+1} (order matters), some edges v_j = x_j - x_1 that make matrix V = (v_1, \dots, v_n) and \sigma_+(S) = \max_j \lVert...
  4. FourEyedRaven

    Netflix series DARK - Taking Sci-Fi Writing to a New Level

    Hi. Any fans of DARK here? This show was mind blowing. The feel of the show may be to heavy for some, especially in the beginning. So it may not be your style. But if you like that atmosphere, the series is truly amazing. The writing is on a whole new level. If you haven't watched it, I...
  5. C

    MHB Using Cauchy Integral Formula for Laurent Series Coefficients

    Dear Everyone, I am wondering how to use the integral formula for a holomorphic function at all points except a point that does not exist in function's analyticity. For instance, Let $f$ be defined as $$f(z)=\frac{z}{e^z-i}$$. $f$ is holomorphic everywhere except for $z_n=i\pi/2+2ni\pi$ for...
  6. jedishrfu

    Awesome Sci Fi Series that’s hard to explain

    I just watched season one of Tales from the Loop. It has the mood of Interstellar on the Earth where people live quiet lives of desperation. There’s an underground physics lab nicknamed the Loop where the impossible becomes possible. There’s the people whose lives are affected in strange ways...
  7. M

    I What is the Function for the Value of a Convergent Series Sum?

    ##\sum_n \frac{1}{n^c}## converges for ##c\gt 1##. Is there an expression for the value of the sum as a function of ##c##?
  8. agnimusayoti

    Fourier series for trigonometric absolute value function

    First, I try to define the function in the figure above: ##V(t)=100\left[sin(120\{pi}\right]##. Then, I use the fact that absolute value function is an even function, so only Fourier series only contain cosine terms. In other words, ##b_n = 0## Next, I want to determine Fourier coefficient...
  9. Adesh

    Analysis Books for learning Fourier series expansion

    Hello Everyone! I want to learn about Fourier series (not Fourier transform), that is approximating a continuous periodic function with something like this ##a_0 \sum_{n=1}^{\infty} (a_n \cos nt + b_n \sin nt)##. I tried some videos and lecture notes that I could find with a google search but...
  10. Leo Liu

    I How Does the Ratio Test Determine Convergence in Power Series?

    I tried to use the ratio test, but I am stuck on finding the range of the limit. $$\because \left|x-1\right|<1.5=Radius$$ $$\therefore -0.5<x<2.5$$ $$\lim _{n \to \infty} \left| \frac{A_{n+1}(x-1)^{n+1}}{A_n(x-1)^n} \right|$$ $$\lim_{n \to \infty} \frac{A_{n+1} \left|x-1\right|}{A_n} <1$$...
  11. PainterGuy

    I Questions about this video on Taylor series

    Hi, I was watching a video on the origin of Taylor Series shown at the bottom. Question 1: The following screenshot was taken at 2:06. The following is said between 01:56 - 02:05: Halley gives these two sets of equations for finding nth roots which we can generalize coming up with one...
  12. WMDhamnekar

    MHB Solve Math Problem: Mixing Milk & Water to Get 50% Milk

    Hi, A person has 40 litres of milk. As soon as he sells half a litre, he mixes the remainder with half a litre of water. How often can he repeat the process, before the amount of milk in the mixture is 50% of the whole? Detailed explanation is appreciated.:) Solution: I am working on...
  13. JD_PM

    Series expansion of the Lorentz Transformation

    a) I think I got this one (I have to thank samalkhaiat and PeroK for helping me with the training in LTs :) ) $$\eta_{\mu\nu}\Big(\delta^{\mu}_{\rho} + \epsilon^{\mu}_{ \ \ \rho} +\frac{1}{2!} \epsilon^{\mu}_{ \ \ \lambda}\epsilon^{\lambda}_{ \ \ \rho}+ \ ...\Big)\Big(\delta^{\nu}_{\sigma} +...
  14. Adesh

    Capacitors connected in series: Why is the voltage the same?

    Here is a circuit diagram: . We have three capacitors, with capacitances ##C_1##, ##C_2## and ##C_3##. Plates are labelled as ##A_1, A_2, A_3 ... A_6##. Point P is connected to the positive terminal of the battery and point N is connected to the negative terminal of the...
  15. rachelmaddiee

    Voltage drops in a series circuit

    Known: V source = 30.0 V , R1 = 15.0 W, R2 = 15.0 W, R3 = 15.0 W To determine the current, first find the equivalent resistance. I = Vsource/R and R = RA + RB = Vsource/RA + RB 30.0 V/15.0 W + 15.0 W + 15.0 W = 1.5 A This is as far as I could do the work for this question. I’m having trouble..
  16. N

    I A question about operator power series

    Hi All, I've been going through Shankar's 'Principles of Quantum Mechanics' and I don't quite understand the point the author is trying to make in this exercise. I get that this wavefunction is not a solution to the Schrodinger equation as it is not continuous at the boundaries and neither is...
  17. F

    Applied Integrals, Series and Products

    I recently stumbled upon Gradshteyn, Ryzhik: Table of Integrals, Series, and Products and it is worth recommending for all who have to deal with actual solutions, i.e. especially engineers, physicists and all who are confronted with calculating integrals, series and products.
  18. M

    Fourier series and the shifting property of Fourier transform

    Summary:: If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform. So here's my attempt to this problem so far...
  19. B

    Calculus 1 problems: functions, integrals, series

    Mentor note: Moved from technical section, so is missing the homework template. Im doing some older exams that my professor has provided, but I haven't got the solutions for these. Can someone help confirm that the solutions I've arrived at are correct?
  20. P

    I Difficult Series Battery Problem

    Modern batteries use double-sided anode and cathodes for greater energy density. Series wiring of batteries is typically accomplished by connecting the anode of one cell to the cathode of another. However, can series be accomplished by stacking double-sided anode and cathode alternatingly with...
  21. anemone

    MHB Find the last digit of a series

    What is the last digit of $1+2+\cdots+n$ if the last digit of $1^3+2^3+\cdots+n^3$ is 1?
  22. chwala

    Find the sum of a function given a series

    since the first term is ##g(0)= \frac {1}{3}## & last term is ##g(1)=\frac {4}{6}## it follows that the ##\sum_{0}^1 g(x)##= ##\frac {1}{3}##+##\frac {4}{6}=1## is this correct?
  23. B

    Engineering RLC Series Network: Impedance, Current, Power Factor, Phasor Diagram

    Hello. I have completed the following question. My answer: i) Circuit Impedance Reactance = XL = 2 x pi x F x L = 2 x pi x 50 x 0.15 = 47.12 Ohms Reactance of Capacitor = XC = 1/2 x pi x F x C...
  24. C

    MHB Finding a formula for the multiplication of multiple formal power series

    Dear Everyone, I am having trouble with finding a formula of the multiplication 3 formula power series. \[ \left(\sum_{n=0}^{\infty} a_nx^n \right)\left(\sum_{k=0}^{\infty} b_kx^k \right)\left(\sum_{m=0}^{\infty} c_mx^m \right) \] Work: For the constant term: $a_0b_0c_0$ For The linear...
  25. Eclair_de_XII

    How to prove divergence of harmonic series by eps-delta proof?

    Set ##\epsilon=\frac{1}{2}##. Let ##N\in \mathbb{N}## and choose ##n=N,m=2N##. Then: ##\begin{align*} \left|s_N-s_{2N}\right|&=&\left|\sum_{l=1}^N \frac{1}{l} - \sum_{l=1}^{2N} \frac{1}{l}\right|\\...
  26. A

    Exploring LRC AC Series Circuits

    Hi all, In an LRC AC series circuit, at which frequencies are where you are mainly dumping your generator/current energy into capacitor to create electric fields or into the inductor to create magnetic fields? So, for example, at low frequencies, f --> 0, the impedance of the inductor goes to 0...
  27. F

    Sum of a series from n=1 to infinity of n^2/(2+1/n)^n

    I tried to write it as n^2/2^n (1+1/2n)^n But I am stuck there and don't know what to try next.Thanks for any help in advance!
  28. C

    Manipulating a Laurent Series Equation

    Not really a homework problem, just an equation from my textbook that I do not understand. I can't think of any way to even begin manipulating the right hand side to make it equal the left hand side. Just to confirm equality (thanks to another user for suggestion), I multiplied both sides by of...
  29. Wrichik Basu

    LIGO India is hosting a series of live talks on Youtube

    LIGO India EPO (Education and Public Outreach) team is hosting a series of talks on Youtube. No registration or any formalities; just tune into the LIGO India EPO Youtube channel and you can attend the lectures. Following is the list of upcoming talks: 20th April: Speaker: Prof. Ajith...
  30. agnimusayoti

    Why the series is divergent based on the Preliminary test

    Interestingly, If I neglect the ##(-1)^n## or ##(-1)^{n+1}## then apply preliminary test, I could find the limit. Whether the limit is not equal to zero, as in series number 1 and 2, then I can conclude the series is divergent. But, if the limit is equal to zero, as in series number 3, then I...
  31. agnimusayoti

    Does Changing the Starting Point of a Series Affect Its Sum?

    1. Is it because the initial formula start the series from ##n = 2##? 2. If the initial formula is used, can I find ##S##, which $$S=\lim_{n\to\infty} \frac{2}{n^2-1}=\frac{2}{\infty}=0$$? Why that answer is different if the formula is changed.
  32. archaic

    Finding this power series -- Where is my error?

    I have used ##\sim## but meant ##\sum_{k=0}^\infty## my math homework platform is telling me that this is wrong. I've tried using desmos to test it and it was a perfect match. Any ideas on where I went wrong?
  33. Mircro

    Material between capacitor plates - viewed as two capacitors in series

    This is a second grade high school problem, translated from my native language. I don't have a problem with calculating, but with understanding the concept. There is an instruction with the assignment that says: The capacitor can be viewed as a combination of two capacitors in series with...
  34. Phylosopher

    I Why perturbation theory uses power series?

    I am revising perturbation theory from Griffiths introduction to quantum mechanics. Griffiths uses power series to represent the perturbation in the system due to small change in the Hamiltonian. But I see no justification for it! Other than the fact that it works. I searched on the internet a...
  35. L

    MHB Help with understanding this series

    Can anyone help with this problem. I've tried integral test but seems to be too complicated.
  36. B

    Series of questions about Special Relativity

    https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf The questions start at page 44 Whenever I refer to y, y = gamma. 1.1 This question is primarily deriving LV/C^2? How does 2LV / c^2-v^2 becomes 2Lv / c^2(1-v^2/c^2)1.4 On the solution page it shows fig 1.61 and fig...
  37. J

    The voltages across capacitors in series

    I found total capacitance and inserted the total capacitance and emf of cell in equation CV =Q. However I know that there is a resistor connected so that this accounts for lost volts
  38. W

    I Taylor series and variable substitutions

    I'm currently typing up some notes on topics since I have free time right now, and this question popped into my head. Given a problem as follows: Find the first five terms of the Taylor series about some ##x_0## and describe the largest interval containing ##x_0## in which they are analytic...
  39. J

    What are the Paschen and Lyman series in the electromagnetic spectrum?

    I know that the Lyman series is a series of lines in the ultra-violet. So that means a higher frequency so it will fall on the right of the diagram. And Paschen is infrared. So a lower frequency so on the left side of the diagram?
  40. .Scott

    Harvard Team Discovers 'Hemolithin' Protein Series on Asteroid

    A Harvard teams believe they have found a protein series they call "Hemolithin" in an asteroid. Isotopes and other evidence indicates that it is not from a terrestrial source."Astrobiology Web" link arxiv pdf link
  41. F

    Possible Values of x for Convergence of Power Series

    We transform the series into a power series by a change of variable: y = √(x2+1) We have the following after substituting: ∑(2nyn/(3n+n3)) We use the ratio test: ρn = |(2n+1yn+1/(3n+1+(n+1)3)/(2nyn/(3n+n3)| = |(3n+n3)2y/(3n+1+(n+1)3)| ρ = |(3∞+∞3)2y/(3∞+1+(∞+1)3)| = |2y| |2y| < 1 |y| = 1/2...
  42. lottotlyl

    Engineering Why Multiply by Exponential Terms in Fourier Series Calculations?

    i tired using complex identity equation for sin(pi*k/3) but it doesn't work out
  43. F

    Find the interval of convergence of this power series

    ∑((√(x2+1))n22/(3n+n3)) We use the ratio test: ρn = |2(3n+n3)√(x2+1)/(3n+1+(n+1)3)| ρ = |2√(x2+1)| ρ < 1 |2√(x2+1)| < 1 No "x" satisfies this expression, so I conclude the series doesn't converge for any "x". However the answer in the book says the series converges for |x| < √(5)/2. What am...
  44. D

    Does This Series Converge for Different Values of Alpha?

    The series ##\sum_{n=0}^\infty \left( ne^{\frac 3 n}-n \right) \left ( \sin \frac {\alpha} {n} - \frac 5 n\right)## i did ##\sum_{n=0}^\infty n\left( e^{\frac 3 n}-1 \right) \left ( \sin \frac {\alpha} {n} - \frac 5 n\right)## for n going to infinity ## \left( e^{\frac 3 n}-1 \right)##...
  45. F

    Find the Interval of Convergence of this Power Series: ∑(x^2n/(2^nn^2))

    ∑(x2n/(2nn2)) We use the ratio test: ρn = |(x2n2/(2(n+1)2)| ρ = |x2/2| ρ < 1 |x2| < 2 |x| = √(2) We investigate the endpoints: x = 2: ∑(4n/(2nn2) = ∑(2n/n2)) We use the preliminary test: limn→∞ 2n/n2 = ∞ Since the numerator is greater than the denominator. Therefore, x = 2 shouldn't be...
  46. Monoxdifly

    MHB What is the number below 25 in this sequence?

    My partner asked me about questions no. 8 and 9. Number 8 asks about what is the area of the quadrilateral. Number 9 asks about what number is below the number 25. Those are questions for Elementary School Math Olympiads in my country but both of us were having a hard time figuring them out...
  47. Shubhan672

    Power series: x^3/3 + x^9/9 + x^15/15 .......

    Moved from technical forum section, so missing the homework template Let x be a real number. Find the function whose power series is represented as follows: x^3/3! + x^9/9! + x^15/15! ... I see that there is a connection to the power series expansion of e^x but am having difficulty finding...
  48. E

    Putting internal resistance in series with a voltage source

    Suppose I have a galvanic cell, where I've arbitrarily set the (-) anode to have a potential of zero volts and the (+) cathode to ##\epsilon## V. The electrodes are connected via the load, but also via the solutions and salt bridge in the centre. Edit: The two trailing wires are connected to a...
  49. D

    Verify the convergence or divergence of a power series

    At the exam i had this power series but couldn't solve it ##\sum_{k=0}^\infty (-1)^\left(k+1\right) \frac {k} {log(k+1)} (2x-1)^k## i did apply the ratio test (lets put aside for the moment (2x-1)^k ) to the series ##\sum_{k=0}^\infty \frac {k} {log(k+1)}## in order to see to what this...
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