What is Series: Definition and 998 Discussions

In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715.
If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).

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  1. T

    Is this series divergent or convergent?

    Homework Statement ##\sum_{n=1}^{\infty }1+(-1)^{n+1} i^{2n}## Is this series divergent or convergent? Homework Equations 3. The Attempt at a Solution [/B] I tried using the divergent test by taking the limit as ##n## approaches ##{\infty }##, but both ##i^{2n}## and ##(-1)^{n+1}## will...
  2. F

    Voltage Variation in a Series Circuit

    If I had a simple series circuit with only a single resistor, and I used a voltmeter to find the voltage between a point at the end of the circuit and another point, which was moved from the beginning to the end of the circuit, what would I find at these various point? Would the voltage remain...
  3. D

    Calculus II: Convergence of Series with Positive Terms

    Homework Statement https://imgur.com/DUdOYjE The problem (#58) and its solution are posted above. Homework Equations I understand that I can approach this two different ways. The first way being the way shown in the solution, and the second way, which my professor suggested, being a Direct...
  4. Rectifier

    Does this series converge? Using the limit comparison test

    The problem In this problem I am supposed to show that the following series converges by somehow comparing it to ## \frac{1}{k\sqrt{k}} ## : $$ \sum^{\infty}_{k=1} \left( \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} \right) $$ The attempt ## \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} =...
  5. J

    Old 6 volt auto headlight in series with 1154 on 12 volts

    If one hooks a 1154 and an old 6 volt automotive headlamp in series and powers it with 12 volts will the headlamp being that it is higher wattage cause all the current flow thru the smaller wattage bulb causing it to burn out?
  6. E

    Can you help me determine the convergence of these series?

    Homework Statement Determine whether the following series converge, converge conditionally, or converge absolutely. Homework Equations a) Σ(-1)^k×k^3×(5+k)^-2k (where k goes from 1 to infinity) b) ∑sin(2π + kπ)/√k × ln(k) (where k goes from 2 to infinity) c) ∑k×sin(1+k^3)/(k + ln(k))...
  7. DoobleD

    I Fourier series of Dirac comb, complex VS real approaches

    Hello, I tried to compute the Fourier series coefficients for the Dirac comb function. I did it using both the "complex" formula and the "real" formula for the Fourier series, and I got : - complex formula : Cn = 1/T - real formula : a0 = 1/T, an = 2/T, bn = 0 This seems to be valid since it...
  8. Ben Geoffrey

    I Taylor Series Expansion of Quadratic Derivatives: Goldstein Ch. 6, Pg. 240

    Can anyone tell me how if the derivative of n(n') is quadratic the second term in the taylor series expansion given below vanishes. This doubt is from the book Classical Mechanics by Goldstein Chapter 6 page 240 3rd edition. I have attached a screenshot below
  9. S

    LC circuits in series with Diodes

    I need help understanding what will happen when the switch in closed in this circuit. What I want to happen is for Cap B to charge first and then discharge into Cap C. When the charged capacitor begins to discharge, will it charge Caps B and C at the same time? It will have to overcome the...
  10. TheComet

    Series diode and a low power device

    I'm designing a device that consumes 450nA in idle (up to 2mA peak) and the maximum allowed voltage is 3.3V. The power is supplied by a battery with a voltage of 3.6V. One of the problems I've run into is: All LDOs I could find have a quiescent current consumption (Iq) greater than 450nA. The...
  11. J

    Charged capacitors connected in series

    Homework Statement Homework EquationsThe Attempt at a Solution I considered N=2 . Two similar charged capacitors are joined in series i.e positive plate of one is joined to negative of the other . If I consider that there is no movement of charge since both the capacitors are similar and...
  12. F

    Linear algebra matrix to compute series

    Post moved by moderator, so missing the homework template. series ##{a_n}## is define by ##a_1=1 ## , ##a_2=5 ## , ##a_3=1 ##, ##a_{n+3}=a_{n+2}+4a_{n+1}-4a_n ## ( ##n \geq 1 ##). $$\begin{pmatrix}a_{n+3} \\ a_{n+2} \\ a_{n+1} \\ \end{pmatrix}=B\begin{pmatrix}a_{n+2} \\ a_{n+1} \\ a_{n} \\...
  13. M

    A Robustness of time series analysis

    I have a time series model constructed by using ordinary least square (linear). I am supposed to provide some general comments on how one would improve the robustness of the analysis of a time series model (in general). Are there any general advice apart from expanding data, making it more...
  14. D

    Convergence of a series with n-th term defined piecewise

    Homework Statement Test the series for convergence or divergence ##1/2^2-1/3^2+1/2^3-1/3^3+1/2^4-1/3^4+...## Homework Equations rn=abs(an+1/an) The Attempt at a Solution With some effort I was able to figure out the 'n' th tern of the series an = \begin{cases} 2^{-(0.5n+1.5)} & \text{if } n...
  15. M

    MHB What is the name for the function in a series?

    I am having trouble describing the function that I am taking sum of in a series. Like in the example \begin{equation} \sum_{z=0}^{\infty}f(z)\end{equation}: What would I call $f(z)$? Would it be the argument of the series?
  16. L

    Find the Fourier Series of the function

    Homework Statement Find the Fourier series of the function ##f## given by ##f(x) = 1##, ##|x| \geq \frac{\pi}{2}## and ##f(x) = 0##, ##|x| \leq \frac{\pi}{2}## over the interval ##[-\pi, \pi]##. Homework Equations From my lecture notes, the Fourier series is ##f(t) = \frac{a_0}{2}*1 +...
  17. M

    I Which x_0 to use in a Taylor series expansion?

    I already learn to use Taylor series as: f(x) = ∑ fn(x0) / n! (x-x0)n But i don´t see why the serie change when we use differents x0 points. Por example: f(x) = x2 to express Taylor series in x0 = 0 f(x) = f(0) + f(0) (x-0) + ... = 0 due to f(0) = (0)2 to x0=1 the series are...
  18. M

    MHB Divisibility of Terms in an Arithmetic Series

    Arithmetic Series? Given the arithmetic series 5+14+23+...(to 241 terms), how many terms in the series are divisible by 5? I need a good explanation and a good start.
  19. GANESH SHETTI

    One 160 kW motor vs two 129 kW motors

    This is related to steel wire pulling machine. I have 2 cases. In CASE-1: ONE MOTOR 160 kW AND CASE-2: TWO MOTORS 129 kW Each. i.e 2x129 kW Which case would have an efficient pulling force and efficiency Case 1 or Case 2.
  20. U

    Determining whether the series is convergent or divergent

    Homework Statement Determine if the series is convergent. Homework Equations ∞ ∑ (((2n^2 + 1)^2)*4^n)/(2(n!)) n=1[/B] The Attempt at a Solution I'n using the Ratio Test and have got as far as (4*(2(n+1)^2+1)^2)/((n+1)((2n^2+1)^2)). I know this series converges but I need to find the...
  21. A

    Trouble determining the Fourier Cosine series for a Function

    Homework Statement I am only interested in 9 (a) Determine the Fourier Cosine series of the function g(x) = x(L-x) for 0 < x < L Homework Equations The Answer for 9 a. g(x) = (L^2)/6 - ∑(L^2/(nπ)^2)cos(2nπx/L) This is the relevant equation given where ω=π/L f(t) = a0+∑ancos(nωt) a0=1/L...
  22. rocky4920

    FInding current in parallel and series circuits

    Homework Statement a. What is the net resistance in the circuit? b. What is the current through the 4 Ω resistor? c. What is the voltage drop across the 3 Ω resistor? Homework Equations R12 = R1 +R2 R34 =R3 + R4 1/Rt = 1/R12 + 1/R34 I3=I4 The Attempt at a Solution a) R12 = 2+ 3...
  23. Euler2718

    Limit of Partial Sums involving Summation of a Product

    Homework Statement Show that the sequence of partial sums s_{n} = 1+\sum_{i=1}^{n} \left(\prod_{k=1}^{i}\left( \frac{1}{2} + \frac{1}{k}\right)\right) converges, with n\in \mathbb{N}\cup \{0\} Homework EquationsThe Attempt at a Solution [/B] So we want to find \lim_{n\to\infty} s_{n} =...
  24. ZapperZ

    B MinutePhysics Special Relativity Series

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  25. F

    Circuits question, series vs parallel

    Homework Statement There are 2 circuits. A: -A series circuit Components: -Motor -Filament lamp -Resistor B: -A parallel circuit Components: -Motor -Filament lamp -Resistor -Each component is in a separate parallel circuit Question)Explain why the power of the motor is lower in the circuit...
  26. J

    Series Solution to Second Order DE

    Homework Statement Consider a power series solution about x0 = 0 for the differential equation y'' + xy' + 2y = 0. a) Find the recurrence relations satisfied by the coefficients an of the power series solution. b) Find the terms a2, a3, a4, a5, a6, a7, a8 of this power series in terms of the...
  27. lfdahl

    MHB Find the exact sum of the series 1/(1⋅2⋅3⋅4)+1/(5⋅6⋅7⋅8)+....

    Find the exact sum of the series: $$S = \frac{1}{1\cdot 2\cdot 3\cdot 4}+\frac{1}{5 \cdot 6 \cdot 7 \cdot 8}+...$$
  28. nmsurobert

    I Balmer Series Lines: How Do Hot Stars Contain Hydrogen?

    Im reading that very hot stars and very cool stars have weak hydrogen lines. With that being said, how do we know that these very hot stars contain high quantity of hydrogen if we can't see it in the spectra?
  29. A

    Laurent series of z^2sin(1/(z-1))

    Homework Statement Find Laurent series of $$z^2sin(\frac{1}{1-z})$$ at $$0<\lvert z-1 \rvert<\infty$$ Homework Equations sine series expansion. The Attempt at a Solution At first, it seems pretty elementary since you can set w=\frac{1}{z-1} and expand at infinity in z, which is 0 in w...
  30. A

    I Complex Fourier Series: Even/Odd Half Range Expansion

    Does the complex form of Fourier series assume even or odd half range expansion?
  31. C

    Coefficient Matching for different series

    Homework Statement Hello, I have a general question regarding to coefficient matching when spanning some function, say , f(x) as a linear combination of some other basis functions belonging to real Hilbert space. Homework Equations - Knowledge of power series, polynomials, Legenedre...
  32. Richie Smash

    Positioning of Resistors in Series

    Hi, I'm aware that the total resistance in a series connection is the sum of all the resistors involved, and that the current is the same throughout, and that the voltage will be different for each resistor but the total voltage will be their sum as well. However, I would like to inquire, does...
  33. N

    Power Series Equation for Amplifier and Harmonics

    Hi, I keep reading in multiple sources that amplifier output can be given by Vout = a0 + a1v(t) + a2v2(t) + a3v3(t) + ... + anvn(t) I've checked in three of my textbooks and there is not a clear definition (its often just stated) why this equation is used and why it works. I am not looking...
  34. M

    MHB Proving Limits of Exponential Series at Infinity

    Hey! :o I want to show that $\displaystyle{\lim_{x\rightarrow \infty}\frac{e^x}{x^{\alpha}}=\infty}$ and $\displaystyle{\lim_{x\rightarrow \infty}x^{\alpha}e^{-x}=0}$ using the exponential series (for a fixed $\alpha\in \mathbb{R}$). I have done the following: $$\lim_{x\rightarrow...
  35. G

    I Classifying Series Summation $$ \sum_{i=0}^{n} 2^{2^i} ~ ?$$

    I am asking on the spur, so there has not been too much thought put into it, but how would we classify a series summation such as $$ \sum_{i=0}^{n} 2^{2^i} ~ ?$$ It does not feel to be geometric, nor that it can be made to be geometric. In general, the function xx does not look like it bears a...
  36. Duke Le

    I [Signal and system] Function with fourier series a[k] = 1

    We have: Period T = 4, so fundamental frequency w0 = pi/2. This question seems sooo easy. But when I use the integral: x(t) = Σa[k] * exp(i*k*pi/2*t). I get 1 + sum(cos(k*pi/2*t)), which does not converge. Where did I went wrong ? Thanks a lot for your help.
  37. E

    MHB Calculate Limit of Series: Step-by-Step Guide

    hey I am trying to calculate the limit of : limn→∞(1/2+3/4+5/8+...+2n−1/2^n) but I am not sure how to solve it, I thought to calculate 2S and than subtract S, but it did not worked well. I did noticed that the denominator is a geometric serie,but I don't know how to continue. could you help?
  38. S

    Fourier Series Expansion

    Homework Statement There is a sawtooth function with u(t)=t-π. Find the Fourier Series expansion in the form of a0 + ∑αkcos(kt) + βksin(kt) Homework Equations a0 = ... αk = ... βk = ... The Attempt at a Solution After solving for a0, ak, and bk, I found that a0=0, ak=0, and bk=-2/k...
  39. Arman777

    Series Comparison Test for Divergence: sin(1/n) vs 1/(1+n)

    Homework Statement ##\sum _{n=0}^{\infty }\:\sin \left(\frac{1}{n}\right)## Homework Equations The Attempt at a Solution Can I try comparison test by ##\left(\frac{1}{1+n}\right)<sin\left(\frac{1}{n}\right)## since ##\left(\frac{1}{1+n}\right)## diverges also...
  40. M

    MHB Can we somehow modify the Lagrange form to get a tighter bound? (Curious)

    Hey! :o I am looking at the following: Show that $\displaystyle{\text{exp}(1)=\sum_{k=0}^{\infty}\frac{1}{k!}=e}$ with $\displaystyle{e:=\lim_{n\rightarrow \infty}\left (1+\frac{1}{n}\right )^n}$. Hint: Use the binomial theorem and compare with the partial sum $s_n$ of the series...
  41. Tony Meloni

    Question about Full scale deflection

    Homework Statement hello, just came across this type of question for first time. A voltmeter with a range of 0-30volts is to be used to measure a 120 volt circuit. calculate the value of the resistor to be placed in series with the meter. the sensitivity of the meter is 1000 ohms per volt ...
  42. J

    SHM Basics -- Series and parallel springs, conceptual question

    Suppose we are asked to find the time period of vertical oscillations of this system. Then should we find the component of displacement along each spring and then add the forces by vector method or should we simplify the diagram into series and parallel connection like in electrical circuits and...
  43. PainterGuy

    How to represent a periodic function using Taylor series

    Hi, Is this possible to represent a periodic function like a triangular wave or square wave using a Taylor series? A triangular wave could be represented as f(x)=|x|=x 0<x<π or f(x)=|x|=-x -π<x<0. I don't see any way of doing although I know that trigonometric series could be used instead...
  44. ElPimiento

    Coefficients for an exponential Fourier Series

    I'm kinda just hoping someone can look over my work and tell me if I'm solving the problem correctly. Since my final answer is very messy, I don't trust it. 1. Homework Statement We're asked to find the Fourier series for the following function: $$ f(\theta)=e^{−\alpha \lvert \theta \rvert}}...
  45. M

    Practice exam question on series

    Homework Statement Find the value of the sum (Infinity) Σ 2/((n+1)(n+3)) (n=1) Homework Equations Integral test Partial Sum Formula = k/2 (a_1 + a_k) The Attempt at a Solution Admittedly I started off this problem the wrong way. I used the integral test thinking I might get an answer...
  46. T

    MHB Power Series Convergence Assistance

    The power series $$\sum_{n = 2}^\infty \frac{(n-1)(-1)^n}{n!}$$ converges to what number? So far, I've tried using the Ratio Test and the limit as n approaches infinity equals $0$. Also since $L<1$, the power series converges by the Ratio Test.
  47. R

    Proving the convergence of series

    Homework Statement Prove the convergence of this series using the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is: The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞ The question is also attached as a .png file 2. Homework Equations The...
  48. maistral

    Zener Model in Series: Applications in Chemical Engineering

    While I was studying applications of Laplace transforms this thing showed up (lol). I seem to have a basic understanding of how the Zener model was derived. Seeing that the time-domain model apparently looks like a step function of sorts, I was trying to relate it to something like the behavior...
  49. M

    Capacitors in Series: Intuitive Understanding Question

    I understand algebraically that when capacitors are in series, the total capacitance is less than any individual capacitance, but I do not understand this intuitively. How can this be possible? Shouldn't more capacitors equal more capacitance?
  50. J6204

    What formula should be used to find the Fourier series of an even function?

    Homework Statement In the following problem I am trying to extend the function $$f(x) = x $$ defined on the interval $$(0,\pi)$$ into the interval $$(-\pi,0)$$ as a even function. Then I need to find the Fourier series of this function.Homework EquationsThe Attempt at a Solution So I believe I...
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