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. S

    Measuring Voltage across Resistor in Series RLC Circuit

    I was experimenting with resonant frequency of a series RLC circuit: 5V AC source 10 ohms resistor 100microF capacitor 46mH inductor Resonant frequency is calculated to be around 74.2Hz. So I set the AC source to resonant frequency 74.2Hz and measured the voltage across the 10 ohms resistor...
  2. I

    MHB Find Limit of Series w/o Quotations: 65 Characters

    Let {a}_{n+1} = \frac{4}{7}{a}_{n} + \frac{3}{7}{a}_{n-1} where a0 = 1, and a1 = 2. Find \lim_{{n}\to{\infty}}{a}_{n} Well, seeing as it says that x approaches infinity, the difference between where points an-1, an, and an+1 are plotted on the y-axis is almost insignificant, so we can simply...
  3. DetectiveT

    Gamma and x-ray decay in different series

    Is there any gamma decay or x-ray decay in Actinium series, Uranium series or Thorium series? On Wikipedia, it only shows alpha and beta decay, does it mean high energy photon decay (gamma or x-ray) exists in each process? Thank you! https://en.wikipedia.org/wiki/Decay_chain#Actinium_series
  4. S

    MHB Discounting a finite series of costs at unknown times

    Consider a finite series of repeated costs C that occur at a series of times ti. Is there a solution to discount these costs by interest rate r to account for time value of money, i.e. solve for S? The times ti of each cost are unknown, but the number of costs n is known, and the average time...
  5. TheSodesa

    A difficult series expansion (finding a limit)

    Homework Statement Find \lim_{x \to 0}\frac{ln(1+x^2)}{1-cos(x)} by using series representations. Check using L'Hospitals rule. Homework Equations Taylor polynomial at x=0: \sum_{k=0}^{\infty}\frac{f^{k}(0)}{k!}(x)^{k} = f(0) + f'(0)(x) + f''(0)x^{2} +... The Attempt at a Solution Using...
  6. M

    Approximate a spectrum from a series of measurements

    Hi. I am working on a linear algebra problem that arose somewhat like this: Suppose that you are shining a light with a known intensity spectrum P(\lambda) upon a surface with an unknown reflection spectrum, R(\lambda). You have a detector to detect the total reflected light intensity, I. How to...
  7. I

    MHB Sum of an Infinite Arithmetic Series

    Somewhere I saw that the sum of the infinite arithmetic series \sum_{n=1}^{\infty}n = \frac{-1}{12} Why exactly is this? I thought infinite arithmetic series had no solution? Also... WHY is it negative? Seems counter-intuitive that the sum of all the NATURAL numbers is a decimal, a negative...
  8. Imtiaz Ahmad

    Current in Series: Why Does It Remain Same?

    hey, sir i have a question why the current remain same in series combination of resistance?
  9. RJLiberator

    Help solving fourier cosine series related problem

    Homework Statement I am doing #9. Homework EquationsThe Attempt at a Solution I've been looking at a lot of similar problems on the internet. The main difference between this one and them is that this one has an interval of [0,4] while they often have intervals of [0,pi] or [-pi,pi] In my...
  10. RJLiberator

    Fourier Series and deriving formulas for sums of numerical

    Homework Statement Homework EquationsThe Attempt at a Solution So I am tasked with answer #3 and #4. I have supplied the indicated parenthesis of 8 also with the image. Here is my thinking: Take the Fourier series for |sin(θ)|. Let θ = 0 and we see a perfect relationship. sin(0) = 0 and...
  11. T

    Finite geometric series formula derivation? why r*S?

    what is the rationale of multiplying "r" to the second line of series? why does cancelling those terms give us a VALID, sound, logical answer? please help. here's a video of the procedure
  12. Euler2718

    Showing the sum of this telescoping series

    Homework Statement Determine whether each of the following series is convergent or divergent. If the series is convergent, find its sum \sum_{i=1}^{\infty} \frac{6}{9i^{2}+6i-8} Homework Equations Partial fraction decomposition \frac{1}{3i-2} - \frac{1}{3i+4} The Attempt at a Solution...
  13. evinda

    MHB Proving Series Convergence: Comparing $\sum y_n$ with $\sum \frac{y_n}{1+y_n}$

    Hello! (Wave) We have a sequence $(y_n)$ with $y_n \geq 0$. We assume that the series $\sum_{n=1}^{\infty} \frac{y_n}{1+y_n}$ converges. How can we show that the series $\sum_{n=1}^{\infty} y_n$ converges? It holds that $y_n \geq \frac{y_n}{1+y_n}$. If we would have to prove the converse we...
  14. Hepth

    Puiseux/Taylor Expansion of an Integrand pre-Integration

    My problem : I have a function that I want to integrate, in the limit that a parameter goes to zero. I have a function ##f[x,r]## I want to compute ##F[r] = \int dx f[x,r]## and then series expand as ##r \rightarrow 0## This is impossible algebraically for me, but may be possible if I can...
  15. Daniel Lobo

    Finding a limit using power series expansion

    Homework Statement The problem wants me to find the limit below using series expansion. ##\lim_{x \to 0}(\frac{1}{x^2}\cdot \frac{\cos x}{(\sin x)^2})## Homework EquationsThe Attempt at a Solution (1) For startes I'll group the two fractions inside the limit together ##\lim_{x...
  16. I

    Checking Taylor Series Result of 6x^3-3x^2+4x+5

    Homework Statement Use zero- through third-order Taylor series expansion f(x) = 6x3 − 3x2 + 4x + 5 Using x0=1 and h =1. Once I found that the Taylor Series value is 49. I want to be able to check the value. On the board our teacher plugged in a value into the equation to show that the answer...
  17. B

    Is there an easier way to do this question about series?

    Hey guys, the question is 6.b. in the picture : http://imgur.com/FaKUMUZ Here is what I did to solve it : http://imgur.com/YrIvbTO I made these two simultaneous equations. 1875 comes from the fact that U1 + U2 = 1500 and U3 + U4 = 375. Then S4 must equal 1500+ 375(1875). I then found a formula...
  18. D

    Series expansion for 2D dipole displaced from the origin

    I learn that we can expand the electric potential in an infinite series of rho and cos(n*phi) when solving the Laplace equation in polar coordinates. The problem I want to consider is the expansion for the potential due to a 2D line dipole (two infinitely-long line charge separated by a small...
  19. F

    Infinite series as the limit of its sequence of partial sums

    In my book, applied analysis by john hunter it gives me a strange way of stating an infinite sum that I'm still trying to understand because in my calculus books it was never described this way. It says: We can use the definition of the convergence of a sequence to define the sum of an...
  20. ognik

    MHB Integrating Fouries series problem

    As the 2nd part of a question, we start with the Fourier sin series expansion of dirac delta function $\delta(x-a)$ in the half-interval (0,L), (0 < a < L): $ \delta(x-a) = \frac{2}{L} \sum_{n=1}^{\infty} sin \frac{n \pi a}{L} sin \frac{n \pi x}{L} $ The questions goes on "By integrating both...
  21. ognik

    MHB Please help me find Fourier series mistake

    Find the Fourier sin series expansion of dirac delta function $\delta(x-a)$ in the half-interval (0,L), (0 < a < L): Now $b_n = \frac{1}{L} \int_0^L f(x)sin \frac{n \pi x}{L}dx $ - but L should be $\frac{L}{2}$ for this exercise... So I would get $ \frac{2}{L} \int_0^L f(x)sin \frac{n \pi...
  22. P

    Difference between lights connected in series and parallel

    Homework Statement When two same lamps are connected with the same battery. Their lighting will be greater when they are connected in series or parallel? Homework Equations Series U=U1+U2+U3+... I=I1=I2=I3... Parallel U=U1=U2=U3... I=I1+I2+I3+... The Attempt at a Solution The answer is when...
  23. I

    Convergence of alternating series

    Homework Statement Do the following series converge or diverge? ## \sum_{n=2}^\infty \frac{1}{\sqrt{n} +(-1)^nn}## and ##\sum_{n=2}^\infty \frac{1}{1+(-1)^n\sqrt{n}}##. Homework Equations Leibniz convergence criteria: If ##\{a_n\}_{k=1}^\infty## is positive, decreasing and ##a_n \to 0##, the...
  24. I

    Proof that e is irrational using Taylor series

    Homework Statement Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number. Use this to show that ##e## is irrational. (Hint: set ##e=p/q## and ##n=q##)...
  25. N

    Radiant power of a series of light waves

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  26. REVIANNA

    Converging Series: Comparison Test w/ 1/n^2

    Mod note: Moved from Homework section I know that ##1/n^4## converges because of comparison test with ##1/n^2## (larger series) converges . how do I know ##1/n^2## converges? coz I cannot compare it with ##1/n## harmonic series as it diverges. @REVIANNA, if you post in the Homework & Coursework...
  27. N

    Charging of capacitors in series

    Hello everyone, I have a doubt about charging of capacitors in series. Suppose I connected two capacitors of same value, say,1 mF in series and put a bulb in series with them and applied a voltage 10V across the series. In steady state, the voltage across each capacitor will be 10/2=5V. Right...
  28. S

    Solving the Fourier Series of a 2π-Periodic Function

    Homework Statement The odd 2π-periodic function f(x) is defined by f(x) = x2 π > x > 0 -x2 −π<x<0 Find the coefficient bn in the Fourier series f(x) = a0/2 + ∑(an cos(nx) + bn sin(nx)). What are the values of the coefficients a0 and an and why? Homework Equations bn = 1/π ∫...
  29. bananabandana

    Euler Lagrange Derivation (Taylor Series)

    Mod note: Moved from Homework section 1. Homework Statement Understand most of the derivation of the E-L just fine, but am confused about the fact that we can somehow Taylor expand ##L## in this way: $$ L\bigg[ y+\alpha\eta(x),y'+\alpha \eta^{'}(x),x\bigg] = L \bigg[ y, y',x\bigg] +...
  30. W

    Series voltage source vs parallel voltage source

    Power systems isn't my area of specialty and I've been doing some reading where it was stated that series voltage connections are safer than parallel connection. I don't fully understand why though. website address...
  31. W

    Taylor polynomial/series, series, function series

    well, i have an calculus exam tomorrow and I'm 100% gona fail. I've neglected calculus so i could study for other subjects and left only 2 days to study taylor's polynomial aproximation, series and function series, the latter two are way more complicated than i expected. good thing is i can...
  32. RealKiller69

    Help with Sum ∑n!/(3*4*5...*n)

    Homework Statement ∑n!/(3*4*5...*n) s1=1/3 sn=1/3+2/(4*3)+3!/(5*4*3)+...+n!/(3*4*5*...n) so i multiplied the sum with 1/2sn=1/6+1/(4*3)+1/(5*4)+1/(6*5)...+1/((n+2)(n-1)) got blocked here,i don't know how to continue, help please
  33. sinkersub

    A: Reciprocal series, B: Laurent Series and Cauchy's Formula

    Problem A now solved! Problem B: I am working with two equations: The first gives me the coefficients for the Laurent Series expansion of a complex function, which is: f(z) = \sum_{n=-\infty}^\infty a_n(z-z_0)^n This first equation for the coefficients is: a_n = \frac{1}{2πi} \oint...
  34. sinkersub

    Inverse Binomial Expansion within Laurent Series?

    Homework Statement Find the Laurent Series of f(z) = \frac{1}{z(z-2)^3} about the singularities z=0 and z=2 (separately). Verify z=0 is a pole of order 1, and z=2 is a pole of order 3. Find residue of f(z) at each pole. Homework Equations The solution starts by parentheses in the form (1 -...
  35. C

    Current Algebra: Find Current in Electric Circuit w/ 3 Resisters of R

    Homework Statement An electric circuit consists of 3 identical resistors of resistance R connected to a cell of emf E and negligible internal resistance. What is the magnitude of the current in the cell? (in the diagram two of the resistors are in parallel with each other then the other in...
  36. Z

    MHB Series Convergence with Comparison Test

    Hey, I am working on Calculus III and Analysis, I really need help with this one problem. I am not even sure where to begin with this problem. I have attached my assignment to this thread and the problem I need help with is A. Thank you!
  37. P

    Find the internal series resistance of the battery

    Homework Statement A battery is connected with a resistor R1=4 om and then it is replaced with the resistance 9 om. In both cases the heat released in the same time is the same. Find the inner resistor of the battery. Homework Equations Q=UIt (U-tension; I-intensity, t-time) I=e.m.f/R+r The...
  38. nomadreid

    Interval of convergence for Taylor series exp of 1/x^2

    Homework Statement The interval of convergence of the Taylor series expansion of 1/x^2, knowing that the interval of convergence of the Taylor series of 1/x centered at 1 is (0,2) Homework Equations If I is the interval of convergence of the expansion of f(x) , and one substitutes a finite...
  39. cnh1995

    Capacitors in series -- microscopic view

    I learned about surface charge feedback theory in electrical circuits a few months ago and it has been extremely helpful for me to intuitively understand many concepts in electrical circuit analysis including conservative and non-conservative fields. I initially referred the paper written by...
  40. C

    Using series in a 2D kinematics problem

    Homework Statement A ball is rolling towards a rectangular hole which is 40cm deep and 2cm wide with a velocity 1m/s. It falls through the hole, bounces off the walls a couple of times and falls down. The direction of balls motion is perpendicular to the hole (falling in it from one side)...
  41. Vinay080

    Other Experimental Researches in Electricity: Series One

    I have already got Faraday's "Experimental Researches in Electricity: Volume 1", which consists of 14 series of experiments concentrated in only one book. But, I wanted to see the more concentrated books, i.e the books of each series, to understand the situation then in elementary way. And even...
  42. ognik

    MHB Is the half interval Fourier series for f(x)=x over (0,L) correct?

    Please help me find my mistake - "find the Sine F/series of f(x)=x over the half-interval (0,L)" I get $ b_n=\frac 2L \int_{0}^{L}x Sin \frac{2n\pi x}{L} \,dx $ $ = \frac 2L \left[ x(-Cos \frac{2n\pi x}{L}. \frac{L}{2n\pi x}\right] + \frac {1}{n\pi} \int_{0}^{L} Cos \frac{2n\pi x}{L} \,dx$...
  43. n.easwaranand

    Zener diode : Calculating the series resistance

    Homework Statement : [/B] Given a voltage regulator with 6.8V Zener diode, input voltage range of 15-20V and load current range 5mA-20mA. Calculate the series resistance R for the regulator.Homework Equations : [/B] Applying KVL and no load situation, we get R = (V - Vz)/Iz where V is the...
  44. 22990atinesh

    Approximating logarithmic series

    Can anybody tell me how this is possible
  45. ognik

    MHB What region for Laurent series

    Please help me with this Laurent series example for $\frac{1}{z(z+2)}$ in the region 1 < |z-1| < 3 Let w = z-1, then $ f(z) = \frac{1}{(w+1)(w+3)}=\frac{1}{2} \left[ \frac{1}{w+1}-\frac{1}{w+3} \right]$ I get $ \frac{1}{1-(-w)} = \sum_{n=0}^{\infty}(-1)^n w^n, \:for\: |w|<1;$ $ = -...
  46. Nono713

    MHB Divergence of a trigonometric series

    Show that this series diverges: $$\sum_{n = 0}^\infty \cos \left ( n^2 \right )$$ (in the sense that it takes arbitrarily large values as $n \to \infty$)
  47. ognik

    MHB Find Fourier series of Dirac delta function

    Hi - firstly should I be concerned that the dirac function is NOT periodic? Either way the problem says expand $\delta(x-t)$ as a Fourier series... I tried $\delta(x-t) = 1, x=t; \delta(x-t) =0, x \ne t , -\pi \le t \le \pi$ ... ('1' still delivers the value of a multiplied function at t)...
  48. ognik

    MHB Help with Fourier series mistake

    Hi - frustratingly I get some problems right 1st time, others just defy me (Headbang) $f(x) = -x, [-\pi,0]; = x, [0,\pi]$ I get $a_0 = \pi$ and $a_n = \frac{-4}{\pi \left(2n-1\right)^2}$ which agrees with the book - but I thought I'd check $b_n$ for practice, it should = 0 according to the...
  49. ognik

    MHB Understanding Fourier Series: Solving a Problem with Sinusoidal Functions

    Hi, appreciate some help with this FS problem - $f(t)= 0$ on $[-\pi, 0]$ and $f(t)=sin\omega t$ on $[0,\pi]$ I get $a_0=\frac{2}{\pi}$ and $b_1 = \frac{1}{2}$, which agree with the book; all other $b_n = 0$ because Sin(mx)Sin(nx) orthogonal for $m \ne n$ But $a_n...
  50. ognik

    MHB How to decompose a fourier series

    Hi, in a section on FS, if I were given $\sum_{n=1}^{\infty} \frac{Sin nx}{n} $ I can recognize that as the Sin component of a Fourier Series, with $b_n = \frac{1}{n} = \frac{1}{\pi} \int_{0}^{2 \pi}f(x) Sin nx \,dx$ Can I find the original f(x) from this? Differentiating both sides doesn't...
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