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).

View More On Wikipedia.org
Recent contents Top users
  1. I

    Current in RLC Series Circuit: Need Help

    So am trying to find the current in the RLC series circuit ,but i think i have done something wrong ,if anyone could tell me where i went wrong ,it would be great ,thank you Resistor-100ohms Capacitor-0.01uF Inductor-25mH Voltage Source-50v a.c 1kHz
  2. O

    I Series for coth(x/2) via Bernoulli numbers

    Hello, I've been using "Guide to Essential Math" by S.M. Blinder from time to time to stay on top of my basic mathematics. I'm currently on the section on Bernoulli Numbers. In that section he has the following (snippet below). Is the transition to equation 7.61 just wrong? The equation just...
  3. S

    MHB Fourier Series involving Hyperbolic Functions

    Hello everyone first time here. don't know if it's the correct group... Am having some issues wiz my maths homework that going to count as a final assessment. Really Really need help. The function (f), with a period of 2π is : f(x) = cosh(x-2π) if x [π;3π].. I had to do a graph as the first...
  4. F

    MHB Prove that series converges uniformly

    I was studying uniform convergence. I have doubts a) Prove that series $\displaystyle\sum_{n=1}^\infty{\displaystyle\frac{ln(1+nx)}{nx^n}}$ converges uniformly on the set $ S = [2, \infty) $. b Prove that series $\displaystyle\sum_{n=1}^\infty{(-1)^{n+1} \displaystyle\frac{e^{-nt}}{\sqrt[...
  5. V

    Connecting two charged capacitors in series

    I looked at the solution of this problem since its a solved problem. I am having doubts with the charges relationship as is mentioned in screenshot below. The charges ##{q_3}^{'}## and ##{q_4}^{'}## are the charges after a a state of balance is reached. Why would the charges have the...
  6. V

    Uncharged capacitors connected in series

    I came across the following explanation from the famous book of Sears and Zemansky which I am unable to understand. I can get the initial part where a positive charge goes to the top plate of C1 since the point a is at a +ve potential causing free electrons to transfer from top plate of C1 to...
  7. P

    MHB Coefficent in a infinite power series

    How do I find a coefficent of x^9 in a power series like this:
  8. A

    The value of a Fourier series at a jump point (discontinuity)

    Greetings according to the function we can see that there is a jump at x=e and I know that the value of the function at x=e should be the average between the value of f(x) at this points my problem is the following the limit of f(x) at x=e is -infinity and f(e)=1 how can we deal with such...
  9. Wrichik Basu

    How do you calculate the series resistance for this Triac control circuit?

    Disclaimer: Some of you might easily recognize that the components and circuit I am talking about are related to one of my projects, on which I had posted some months ago. Actually, the circuit is the same as the one in my project, but the one I am posting in this thread actually uses high...
  10. Fochina

    Finding the convergence of a parametric series

    It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with...
  11. A

    A problem with the convergence of a series

    Good day I have a question about the convergence of the following serie I understand that the racine test shows that it an goes to 2/3 which makes it convergent but I also know that for a sequence to be convergent the term an should goes to 0 but the lim(n---->inf) ((2n+100)/(3n+1))^n)=lim...
  12. R

    I Series Solution for 2nd-Order Homogeneous ODE

  13. L

    A Limits of Taylor Series: Is $\sin x=x+O(x^2)$ Correct?

    We sometimes write that \sin x=x+O(x^3) that is correct if \lim_{x \to 0}\frac{\sin x-x}{x^3} is bounded. However is it fine that to write \sin x=x+O(x^2)?
  14. M

    Nonlinear spring made from many different linear springs in series?

    Can I create a nonlinear spring, for a nonlinear oscillator, by putting many linear springs in series and parallel?
  15. Vividly

    Learning New Math Material: Is the 'For Dummies' Series Helpful?

    I have a question. I’m reading the series of the practical man which include arithmetic, algebra,geometry, Trigonometry and Calculus. I’m having some trouble understanding these books and was thinking about reading the series of ____ for dummies such as Geometry for dummies as a supplement. Is...
  16. A

    I Proving the Finite Binomial Series for k Non-Negative Integer

    Hello, I was wondering how to prove that the Binomial Series is not infinite when k is a non-negative integer. I really don't understand how we can prove this. Do you have any examples that can show that there is a finite number when k is a non-negative integer? Thank you!
  17. Stonestreecty

    Problem to divide two 16 bit numbers in 8051 series microcontrollers

    Hi all, I have a project to code in 8051 series, DS80C320-ECG (data source as reference): "Division of two 16 bit unsigned integers being in the internal memory, quotient and remainder should be stored". I find a way to do it but there is a part of the program that i don't understand, I attach...
  18. T

    B Difficulty with understanding whether 1/n converges or diverges

    Hi, I have a quick question about whether or not the infinite series of 1/n converges or diverges. My textbook tells me that it diverges, but my textbook also says that by the nth term test if we take the limit from n to infinity of a series, if the limit value is not equal to zero the series...
  19. docnet

    I The precise relationship between Fourier series and Fourier transform

    Would someone be able to explain like I am five years old, what is the precise relationship between Fourier series and Fourier transform? Could someone maybe offer a concrete example that clearly illustrates the relationship between the two? I found an old thread that discusses this, but I...
  20. A

    How is this a telescoping series?

    Hello ! Consider this series; $$ \sum_{k=1}^{\infty} \frac{1}{(2k-1)(2k+1)} $$ It is said to find the limit of the series when approaches infinity.Now it is said that this is a telescopic series and that the limit is ##\frac{1}{2}## but I don't see it. I've split the an part (I don't know how...
  21. A

    Is Using the Comparison Test to Prove Divergence Valid for This Series?

    Summary:: I am suspossed to find the limit of this series.I've come to realize that the series diverges and I'm trying to prove that using the a comparison test. Hello! Consider this sum $$ \sum_{k=1}^{n} (\sqrt{1+k} - \sqrt{k}) $$ the question wants me to find the limit of this sum where n...
  22. sodoyle

    Can I series buck converters to get higher stepdown ratio?

    I have two capacitors in series across my bus. I have a some series resistors across the capacitors for voltage balancing. I would like to power some low voltage low voltage components tapping off of the voltage divider using that as the voltage supply. The bus voltage will be approximately 2...
  23. stevendaryl

    Insights Computing the Riemann Zeta Function Using Fourier Series

    Continue reading...
  24. Butterfly41398

    Series RLC Circuits: Resonant Frequency of 60° at 40 Hz

    Summary:: About resonant frequencies A series RLC circuit with R = 250 ohms and L = 0.6 H results in a leading phase angle of 60° at a frequency of 40 Hz. At what frequency will the circuit resonate? Answer is 81.2 Hz but i got a different answer. May someone please correct me.
  25. T

    I Question on Harmonic Oscillator Series Derivation

    Good afternoon all, On page 51 of David Griffith's 'Introduction to Quantum Mechanics', 2nd ed., there's a discussion involving the alternate method to getting at the energy levels of the harmonic oscillator. I'm filling in all the steps between the equations on my own, and I have a question...
  26. MathematicalPhysicist

    Mathematica Series expansion from the red book on special functions by Richard Ask

    I want to check my calculations via mathematica. In the book I am reading there's this expansion: $$\frac{(1+\frac{1}{j})^x}{1+x/j}=1+\frac{x(x-1)}{2j^2}+\mathcal{O}(1/j^3)$$ though I get instead of the term ##\frac{x(x-1)}{2j^2}## in the rhs the term: ##-\frac{x(x+1)}{2j^2}##. So I want to...
  27. L

    A Can falling factorials be a Schauder basis for formal power series?

    We usually talk about ##F[[x]]##, the set of formal power series with coefficients in ##F##, as a topological ring. But we can also view it as a topological vector space over ##F## where ##F## is endowed with the discrete topology. And viewed in this way, ##\{x^n:n\in\mathbb{N}\}## is a...
  28. mcas

    Find change in entropy for a system with a series of reservoirs

    I've calculated the change in the entropy of material after it comes in contact with the reservoir: $$\Delta S_1 = C \int_{T_i+t\Delta T}^{T_i+(t+1)\Delta T} \frac{dT}{T} = C \ln{\frac{T_i+(t+1)\Delta T}{T_i+t\Delta T}}$$ Now I would like to calculate the change in the entropy of the...
  29. wcjy

    Source free RLC series circuit

    Hello, this is my working. My professor did not give any answer key, and thus can I check if I approach the question correctly, and also check if my answer is correct at the same time. for t < 0, V(0-) = V(0+) = 60V I(0) = 60 / 50 = 1.2A When t > 0, $$α = \frac{R}{2L}$$ $$α =...
  30. H

    What is the power factor in a series circuit?

    I calculated in the following and got the correct answer. However, I wonder whether this way is correct or not. Thanks! PR / Pavg = Irms^2 * R / Irms^2*Z = 15 /33.36 = 0.45
  31. H

    Understanding RLC Circuits: Series vs Parallel

    I'm a bit confused with RLC circuit. If in series, IL = IL, max * cos(wt) If in parallel, IL = - IL, max * cos(wt) Are these correct?
  32. A

    Problem with the sum of a Fourier series

    Good day I really don't understand how they got this result? for me the sum of the Fourier serie of of f is equal to f(2)=log(3) any help would be highly appreciated! thanks in advance!
  33. S

    MHB Can the Series Sum Be Expressed as an Integral as N Approaches Infinity?

    I wonder if the limit of the following can be converted into integral or some elegant form as N tends to infinity: \[ \sum_{n=0}^{N}\frac{a}{2^{n}}\sin^{2}\left(\frac{a}{2^{n}}\right) \] If we plot or evaluate the value then it does appear that the series converges very fast...
  34. C

    Can you use Taylor Series with mathematical objects other than points?

    I was recently studying the pressure gradient force, and I found it interesting (though this may be incorrect) that you can use a Taylor expansion to pretend that the value of the internal pressure of the fluid does not matter at all, because the internal pressure forces that are a part of the...
  35. A

    Problem in finding the radius of convergence of a series

    Good day I'm trying to find the radius of this serie, and here is the solution I just have problem understanding why 2^(n/2) is little o of 3^(n/3) ? many thanks in advance Best regards!
  36. A

    Studying the convergence of a series with an arctangent of a partial sum

    Greeting I'm trying to study the convergence of this serie I started studying the absolute convergence because an≈n^(2/3) we know that Sn will be divergente S=∝ so arcatn (Sn)≤π/2 and the denominator would be a positive number less than π/2, and because an≈n^(2/3) and we know 1/n^(2/3) >...
  37. hackedagainanda

    Intro Math Schaum's Outline Series, Supplementary Problem books

    I've recently purchased Schaum's Elementary, Intermediate and College Algebra texts and I'm loving the extra practice and the chance to familiarize myself with the material. Are there any other similar books out there for cheap? I've considering buying old editions of the various...
  38. A

    Convergence of a series involving ln() terms in the denominator of a fraction

    good day I want to study the convergence of this serie and want to check my approch I want to procede by asymptotic comparison artgln n ≈pi/2 n+n ln^2 n ≈n ln^2 n and we know that 1/(n ln^2 n ) converge so the initial serie converge many thanks in advance!
  39. A

    Discussing the Convergence of a Series: Get My Opinion!

    Good day I want to study the connvergence of this serie I already have the solution but I want to discuss my approach and get your opinion about it it s clear that n^2+5n+7>n^2+3n+1 so 0<(n^2+3n+1)/(n^2+5n+7)<1 so we can consider this as a geometric serie that converge? many thanks in advance
  40. A

    Problem with series convergence — Taylor expansion of exponential

    Good day and here is the solution, I have questions about I don't understand why when in the taylor expansion of exponential when x goes to infinity x^7 is little o of x ? I could undesrtand if -1<x<1 but not if x tends to infinity? many thanks in advance!
  41. A

    Problem studying the convergence of a series

    Good day here is the exercice and here is the solution that I understand very well but I have a confusion I hope someone can explain me if I take the taylor expansion of sin ((n^2+n+1/(n+1))*pi)≈n^2+n+1/(n+1))*pi≈n*pi which diverge! I know something is wrong in my logic please help me many...
  42. M

    I Regression Prediction with Time Series Data

    Hi, I am not sure what the correct forum is for this question. Question: When do we need to remove seasonality from time series data to do a regression analysis? Context: I am planning to conduct a prediction analysis where I want to find out how a device performs. I hope to estimate a...
  43. F

    Electrical Daisy-chained, parallel or series? (failed outlets in home)

    Hello Forum, Some of my electrical outlets (3) in the kitchen stopped working (one of them is a GFCI outlet). Reading online, I found out that outlets are generally connect in a daisy-chain fashion and if one goes back they all stop working. See the figure below showing a daisy chain...
  44. chwala

    Prove that these two series are equal

    i looked at this and it was not making any sense at all, could it be a textbook error or i am missing something here; note that, lhs gives us, ##4,6,8,10,12,14## rhs gives us, ##8,11,14,17,20,23##
  45. Tony Hau

    Finding the Fourier Series of a step function

    The answer in the textbook writes: $$ f(x) = \frac{1}{4} +\frac{1}{\pi}(\frac{\cos(x)}{1}-\frac{\cos(3x)}{3}+\frac{\cos(5x)}{5} \dots) + \frac{1}{\pi}(\frac{\sin(x)}{1}-\frac{2\sin(2x)}{2}+\frac{\sin(3x)}{3} + \frac{\sin(5x)}{5}\dots)$$ I am ok with the two trigonometric series in the answer...
  46. M

    Engineering Converting a series connected transfer function to the state space model

    Hi, I have a question about a homework problem: I am not sure why I do not seem to get the same answers when using different methods. Question: Given transfer functions G(s) = \frac{s - 1}{s + 4} and C(s) = \frac{1}{s - 1} , find the state space models for those systems. Then find the...
  47. B

    The Other Two Sarah Conners: Uncovering Red Herrings in the Terminator Series

    So, you know that thing with the Wizard of Oz you can't unsee once someone explains how it's actually Glenda the Good Witch who's the real villain? (Glenda blatantly endangered and manipulated Dorothy into killing Glenda's most dangerous enemy, while smiling. Ending up with the Ruby Slippers...
  48. murshid_islam

    I Any Good Lecture Series on Complex Analysis?

    Can anyone suggest a good lecture series on Complex Analysis on YouTube? I have already searched on YouTube myself, and there are a few. But I wanted to know if any of you would recommend some particular lecture series which you consider to be good.
  49. Ivan Seeking

    Great Movies and Series [Not Sci-Fi]

    I have a seen a number of great series about women lately. The Queens Gambit: Loved it! The Marvelous Mrs. Maisel: Of course this has been around since 2013 and just concluded last year Orange is the New Black All on Netflix
  50. asd852

    Circuit analysis involving parallel and series conditions

    par
Back
Top