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. faradayscat

    Differential equation with power series

    Homework Statement Solve y''+(cosx)y=0 with power series (centered at 0) Homework Equations y(x) = Σ anxn The Attempt at a Solution I would just like for someone to check my work: I first computed (cosx)y like this: (cosx)y = (1-x2/2!+x4/4!+ ...)*(a0+a1x+a2x2 +...)...
  2. DevonZA

    Kirchhoff's Law Problem -- Resistors in Series and Parallel and a Switch

    Homework Statement Homework Equations Rparellel=1/R1+1/R2... IR1=R2/R1+R2 x I The Attempt at a Solution I am unsure of how to answer d) and e) using KVL because I count 4 junctions? Where should I start?
  3. Drakkith

    I Geometric Series Convergence and Divergence

    I'm a little confused on geometric series. My book says that a geometric series is a series of the type: n=1 to ∞, ∑arn-1 If r<1 the series converges to a/(1-r), otherwise the series diverges. So let's say we have a series: n=1 to ∞, ∑An, with An = 1/2n An can be re-written as (1/2)n, which...
  4. S

    Convergence of Infinite Series: Solving for the Sum of 1/n^4

    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?
  5. S

    Kirchoff's method for a weird circuit

    Homework Statement In the following circuit, the battery has emf ε = 12.7 V. The resistors are R1 = 2000 Ω . R2 = 3000 Ω, and R3 = 4000 Ω. What is the current through resistor R2 ? Homework Equations Kirchoff's Rules V=IR The Attempt at a Solution i0 (into A) - i2 (current into resistor 2) -...
  6. saybrook1

    How can a -1 exponent be manipulated in the Sinh series?

    Homework Statement Hello, I'm not trying to solve this exact problem although mine is similar and I am confused on how they were able to get a -1 in the exponent from one step to another. Homework Equations I have attached a picture indicating the step that I am confused about. How are they...
  7. henry wang

    I How is the constant pi/L deduced in Fourier series?

    How is pi/L part deduced in (n*pi*x)/L?
  8. Drakkith

    I Alternating Series, Testing for Convergence

    The criteria for testing for convergence with the alternating series test, according to my book, is: Σ(-1)n-1bn With bn>0, bn+1 ≤ bn for all n, and lim n→∞bn = 0. My question is about the criteria. I'm running into several homework problem where bn is not always greater than bn+1, such as the...
  9. K

    Capacitors in series and Parallel

    There are a few problems here but it would be helpful if The below solutions could be checked and some insight provided for the last one/any other mistakes The problem & The Solution Attempts A circuit with a 12V battery then on the row below in series with the battery is a 120nF capacitor...
  10. F

    Harmonic Series: Is ∑(1/ k+1 ) Divergent?

    Homework Statement i know that k = 0 to∞∑(1/ k) is harmonic series( we know that the sum is divergent) , how about ∑(1/ k+1 ) ? Homework EquationsThe Attempt at a Solution in my opinion , it's also harmonic series , because the sum is divergent . Am i right ?
  11. M

    Complex Fourier Series into a Cosine Series

    Homework Statement a. Represent f(x)=|x| in -2<x<2 with a complex Fourier series b. Show that the complex Fourier Series can be rearranged into a cosine series c. Take the derivative of that cosine series. What function does the resulting series represent? [/B]Homework Equations...
  12. ReidMerrill

    Representing a function as a power series

    Homework Statement Represent the function (8x)/(6+x) as a power serioes f(x)=∑cnxn Find c0 c1 c2 c3 c4 Radius of convergence R= Homework EquationsThe Attempt at a Solution I've represented this function as (8x/9)∑(-x/6)n and found I-x/6I <1 so R=6 Through pure guessing I discovered c0=0 but...
  13. R

    Solving ODE with Frobenius Method

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  14. L

    Pointwise, uniform convergence of fourier series

    Hello; I'm struggling with pointwise and uniform convergence, I think that examples are going to help me understand Homework Statement Consider the Fourier sine series of each of the following functions. In this exercise de not compute the coefficients but use the general convergence theorems...
  15. deagledoubleg

    Find Taylor Series from a function and its interval of convergence

    Let f(x) = (1+x)-4 Find the Taylor Series of f centered at x=1 and its interval of convergence. \sum_{n=0}^\infty f^n(c)\frac{(x-c)^n}{n!} is general Taylor series form My attempt I found the first 4 derivatives of f(x) and their values at fn(1). Yet from here I do not know how to find the...
  16. P

    Other Which textbooks can help me achieve my goal of mastering all forms of physics?

    Hey Physics Forums! I am a self taught individual, who would like to learn more about physics. My goal in life is to virtually understand every physics principal we know, and become extremely good at all forms of physics. I will be reading physics books over the next 30 years, so that i can...
  17. The-Mad-Lisper

    Proof for Convergent of Series With Seq. Similar to 1/n

    Homework Statement \sum\limits_{n=1}^{\infty}\frac{n-1}{(n+2)(n+3)} Homework Equations S=\sum\limits_{n=1}^{\infty}a_n (1) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\gt 1\rightarrow S\ is\ divergent (2) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\lt 1\rightarrow S\ is\...
  18. Drakkith

    I Defining Functions as Sums of Series

    My Calculus 2 teacher's lecture slides say: Many of the functions that arise in mathematical physics and chemistry, such as Bessel functions, are defined as sums of series. I was just wondering how this was different from the basic functions that we've already worked with. Are they not...
  19. NihalRi

    Maclaurin series and general calculus question

    Homework Statement This question has four parts which may follow up from each other so I incuded all the parts. The real problem I'm having is with d Consider the function f ang g given by f (x)=( e^x+[e^-x])/2 & g (x) =( [e]^x]-[e^-x])/2 a) show f'(x) = g (x) and g'(x) = f (x) b) find the...
  20. nfcfox

    The power series above is the Taylor series....

    Homework Statement http://imgur.com/1aOFPI7 PART 2 Homework Equations Taylor series form The Attempt at a Solution My thought process is that the answer is 3 because using the geometric series equation (1st term)/(1-R) then you can get the sum. In this case R would be x+2 where x is -2 so 0...
  21. The-Mad-Lisper

    Preliminary Test of Alternating Geometric Series

    Homework Statement \lim_{n \to \infty}\frac{(-1)^{n+1} \cdot n^2}{n^2+1} Homework Equations \lim_{n \to \infty}a_n \neq 0 \rightarrow S \ is \ divergent The Attempt at a Solution I tried L'Hopital's rule, but I could not figure out how to find the limit of that pesky (-1)^{n+1}. Edit: This...
  22. Bigger than smaller

    Conservation of Charge in Series Capacitors.

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  23. Ethan Godden

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  24. M

    MHB Normal series and composition series

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  25. NihalRi

    Use geometric series to write power series representation

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  26. M

    MHB Find all the composition series

    Hey! :o I want to find all the composition series for $A_4$ and $S_3\times\mathbb{Z}_2$. A composition series for $G$ is $$1=S_0\leq S_1\leq S_2\leq \cdots \leq S_k=G$$ with $S_i\trianglelefteq S_{i+1}$ and $S_{i+1}/S_i$ is a simple group, right? (Wondering) Could you give me some hints how...
  27. P

    Series resistors' output voltage

    In my electrical engineering textbook, in the section with voltage dividers, it says that after you combine two series resistors, then the output voltage can no longer be defined. It then said that thus the equivalence was made strictly from a voltage source standpoint. I do not understand why...
  28. F

    Head loss formula in series pipe

    Homework Statement i know that the formula of head loss is (V^2) / 2g , where v =velocity , but , why did the author want to change it to k[(Q)^1/n ] ? Homework EquationsThe Attempt at a Solution
  29. Z

    Infinite series related interest question

    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...
  30. nazmus sakib

    Fourier sine series for a triangular wave on a finite string

    Homework Statement A string of length L =8 is fixed at both ends. It is given a small triangular displacement and released from rest at t=0. Find out Fourier coefficient Bn. Homework Equations what should i use for U0(x) ? The Attempt at a Solution
  31. P

    Classical How good are the MIT introductory physics series?

    I was wondering, are the MIT introductory series any good compared to, let's say, Halliday and Resnick ? Any opninion ?
  32. karush

    MHB Find the sum of the first 17 terms of the arithmetic series

    Find the sum of the first 17 terms of the arithmetic series $$8+\sqrt{7}, \ 6,\ 4-\sqrt{7}$$ $$u=8+\sqrt{7}$$ $$S_{17} =\frac{u\left(1-\frac{{6}^{17}} {u} \right)}{u}$$ My first shot at this
  33. nfcfox

    What Goes in the Maclaurin Series for 2^x?

    Homework Statement Find the Maclaurin series for f(x) by any method. f(x)=2^x Homework Equations d/dx(b^x)= ln(b)b^x The Attempt at a Solution Ok so I basically took the derivative about 3 or so times and came out with ∑ n=0 to ∞ of ((ln(2))^n(something has to go here))/n! This much I have...
  34. Odious Suspect

    I Power Series solution to dy/dx=x+y

    This is from an example in Thomas's Classical Edition. The task is to find a solution to ##\frac{dy}{dx}=x+y## with the initial condition ##x=0; y=1##. He uses what he calls successive approximations. $$y_1 = 1$$ $$\frac{dy_2}{dx}=y_1+x$$ $$\frac{dy_3}{dx}=y_2+x$$ ...
  35. F

    How to Expand Noncommuting Variables in a Formal Power Series?

    Homework Statement Need to show that [a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger} Homework Equations [a,a^\dagger]=1 The Attempt at a Solution Need to expand f(a,a^\dagger) in a formal power series. However I don´t know how to do it if the variables don´t commute.
  36. I

    Parallel and Series Capacitors?

    In the figure below, a potential difference V = 150 V is applied across a capacitor arrangement with capacitances C1 = 12.0µF, C2 = 6.00µF, and C3 = 16.0µF. Find the following values. Here's the diagram: http://www.webassign.net/hrw/hrw7_25-28.gif I already solved this problem but I'm having...
  37. N

    I So I flip 10 coins... (re: limit of infinite? series)

    Originally from the statistics forum but am told this is more of a calculus question. I flip 10 coins, if any of the coins land on tails, all of the coins split into 10 new coins and I flip them all again. I keep doing this until a round where every single coin lands on heads. Can I expect to...
  38. Euler2718

    I Power series Construction Help

    Very basic issue here. Using: \frac{1}{1-x} = \sum_{i=0}^{\infty} x^{i} , |x|<0 Find the power series representation and interval of convergence for: f(x) = \frac{1}{(1-3x)^{2}} We have that: \frac{d}{dx}\left[\frac{1}{1-x}\right] = \frac{1}{(1-x)^{2}} = \sum_{i=0}^{\infty} ix^{i-1} ...
  39. saybrook1

    Trouble with an index change in Laurent series

    Homework Statement Hey guys, I'm just going through a Laurent series example and I'm having trouble understanding how they switched the index on a summation from n=0 to n=1 and then switched the argument from z^(-n-1) to z^n as well as changing the upper limit to -infinity. If anyone could shed...
  40. F

    Inconclusive Results When p=1: Exploring Uk Series

    Homework Statement when the test is inconclusive when p = 1? when p=1 , the sum of uk will grow bigger , to infinity , right ? Homework EquationsThe Attempt at a Solution let's say uk = 2 , uk_2 = 2 , so as uk_3 ,l uk_4 ... the sum of all of them will beocme infinity , right?[/B]
  41. S

    Current flow across inductors in Series

    I can't articulately ask the question so I drew a diagram. In the diagram, both capacitors are equal in capacitance. The bottom inductors are both 10000 nh and the top are 100 nh. A. Will the capacitors charge at the same time? B If we switch the large inductors to the top and the small to the...
  42. C

    Odd or Even? - Arbritrary Period Fourier Series

    Homework Statement Hello everyone, I'm new to the great field that is Fourier analysis, and have a question about the way in which to determine if the function is a odd or even function. Given the function, of one period f(x) = { x; 0 <= x < =1, 1; 1 < x < 2, (3 -x); 2 <= x <= 3: Is...
  43. PhysicsBoyMan

    Find resistance of resistor connected in series

    Homework Statement Two 1.80 V batteries—with their positive terminals in the same direction—are inserted in series into the barrel of a flashlight. One battery has an internal resistance of R1 = 0.280Ω, the other an internal resistance of R2 = 0.155Ω. When the switch is closed, a current of...
  44. RJLiberator

    How to Find Fourier Series for a Given Function Using Sine Series?

    Homework Statement Find the Fourier series for the following function (0 ≤ x ≤ L): y(x) = Ax(L-x) Homework EquationsThe Attempt at a Solution 1. We start with the sum from n to infinity of A_n*sin(n*pi*x/L) where An = B_n*Ax(l-x) 2. We have the integral from 0 to L of f(x)*sin(m*pi*x/L) dx...
  45. J

    How Does Young's Modulus Differ from Spring Constant in Shape and Geometry?

    Homework Statement Use data from your experiment to support the idea that Young's Modulus relates the material and is independent of shape and geometry whilst the spring constant is a function of the shape and geometry. The experiment involved stretching identicle springs (starting off with one...
  46. Crush1986

    How can I find the Laurent series for Cos(1/z) at z=0?

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  47. S

    Series connected zener diodes problem

    I have attached the image with post. Answer to question is given as A. I am not getting the explanation to how can one zener diodes be in breakdown and other not. I think if the combined voltage of 150V is applied then only breakdown occurrs in both the diodes simultaneously.
  48. Euler2718

    How to prove Convergence of this Series

    Homework Statement Use any appropriate test to determine the convergence or divergence of the following series: \sum_{i=0}^{\infty} \frac{2^{i} + 3^{i}}{4^{i}+5^{i}} Homework EquationsThe Attempt at a Solution I've run it through mathematica and it told me it's convergent. However, I...
  49. B

    Solution to series resistance function

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  50. W

    Capacitors in Series and Parallel

    Homework Statement Fig. 25-39 shows a 12.0 V battery and four uncharged capacitors of capacitances C1 = 1.00 µF, C2 = 2.00 µF, C3 = 3.00 µF, and C4 = 4.00 µF. If only switch S1 is closed, what is the charge on (a) capacitor 1, (b) capacitor 2, (c) capacitor 3, and (d) capacitor 4? (figure at...
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