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. StudentOfScience

    I Series for Elliptic Integral of the First Kind

    I'm not sure if this should go in the homework forum or not, but here we go. Hello all, I've been trying to find a series representation for the elliptic integral of the first kind. From some "research", the power series for the complete form (## \varphi=\frac{\pi}{2} ## or ## x=1 ##) seems to...
  2. S

    Finding the sum of a series by grouping

    Homework Statement Homework Equations Summation The Attempt at a Solution I know I could have simplified (3n-2)^3 +(3n-1)^3 -(3n)^3 and put the formulas in but I wonder is there any other method (I was thinking about grouping the terms, but to no avail) to work this out.
  3. rhdinah

    Boas Maclaurin series for ln(2)

  4. doggonemess

    Wanted: specific youtube science series, forgot who made it....

    This is a desperate attempt to find a set of videos I saw about a year ago on YouTube. It was not one of the big, well known guys like Veritasium or SciShow, it was just one middle aged guy. He explained scientific advancements through history, and gave really, really detailed accounts of how...
  5. karush

    MHB Infinite Geometric Series and Convergence

    a. Find the common ration $r$, for an infinite series with an initial term $4$ that converges to a sum of $\displaystyle\frac{16}{3}$ $$\displaystyle S=\frac{a}{1-r} $$ so $\displaystyle\frac{16}{3}=\frac{4}{1-r}$ then $\displaystyle r=\frac{1}{4}$ b. Consider the infinite geometric series...
  6. C

    MHB Series and Convergence Tests

    Hi guys, I am doing this question of alternating series test. And I was following the below principles when solving the problem. Sorry I don't know how to type in the math language. I got 4, 8, 9, 10 as the answers. But the system rejected this without any explanation. Can someone throw a...
  7. karush

    MHB Calculate the sum for the infinite geometric series

    Calculate the sum for the infinite geometric series $4+2+1+\frac{1}{2}+...$ all I know is the ratio is $\frac{1}{2}$ $\displaystyle\sum_{n}^{\infty}a{r}^{n}$ assume this is used
  8. P

    What are the tensor and series questions in this homework?

    Homework Statement i have a few homework question and want to be sure if I have solved them right. Q1) Write ##\vec{\triangledown}\cdot\vec{\triangledown}\times\vec{A}## and ##\vec{\triangledown}\times\vec{\triangledown}\phi## in tensor index notation in ##R^3## Q2) the spherical coordinates...
  9. C

    Sum of sinosoids that can be a Fourier Series expansion

    Homework Statement I was given a problem with a list of sums of sinusoidal signals, such as Example that I made up: x(t)=cos(t)+5sin(5*t). The problem asks if a given expression could be a Fourier expansion. Homework Equations [/B]The Attempt at a Solution My guess is that it has something to...
  10. smodak

    Quantum Found a great book (series) on Quantum Mechanics

    The books are based on Schwinger's but is much easier read. Uses my favorite spins-first approach. Lectures On Quantum Mechanics vol. 1, 2, & 3 by Berthold-Georg Englert https://www.amazon.com/dp/9812569715/?tag=pfamazon01-20 https://www.amazon.com/dp/9812569731/?tag=pfamazon01-20...
  11. M

    Physics Problems on Flash Tv Series

    Hello, i am not sure where to discuss it but here maybe proper for this thread. I just want to discuss about DC's Tv show Flash and physics on it like singularity or parallel universes?
  12. alexandria

    Arithmetic and Geometric Series

    Homework Statement Homework Equations no equations required 3. The Attempt at a Solution a) so for part c) i came up with two formula's for the tortoise series: the first formula (for the toroise series) is Sn = 20n This formula makes sense and agrees with part a). for example, if the...
  13. Y

    Why does putting two batteries in series increase the voltage?

    So say we had 2 batteries, B1 and B2, and B2 is on top of B1. The + terminal of B1 connects to - terminal of B2, and the + terminal of B2 connects to - terminal of B1. Why does this double the voltage compared to the voltage of just B1?
  14. J

    MHB Power Series Problem: Solve f(3x) = 1/(1 - 3x)

    So here is the problem I am trying to solve: You can combine two (or more) convergent power series on the same interval I. Using the properties of the geometric series, find the power series of the function below. Series: f(x) = 1/(1 - x) = sigma k = 0, infinity = 1+ x + x^2 + x^3 Function...
  15. T

    MHB Why is this Maclaurin series incorrect?

    I need to find the Maclaurin series for $$f(x) = x^2e^x$$ I know $$e^x = \sum_{n = 0}^{\infty} \frac{x^n}{n!}$$ So, why can't I do $$x^2 e^x =x^2 \sum_{n = 0}^{\infty} \frac{x^n}{n!} = \sum_{n = 0}^{\infty} \frac{x^2 x^n}{n!} $$
  16. S

    I Linearizing vectors using Taylor Series

    I am linearizing a vector equation using the first order taylor series expansion. I would like to linearize the equation with respect to both the magnitude of the vector and the direction of the vector. Does that mean I will have to treat it as a Taylor expansion about two variables...
  17. M

    Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0

    Homework Statement Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0. Homework Equations None. The Attempt at a Solution I'll post my whole work.
  18. R

    B Before the big bang video series

    I discovered this interesting series of videos that others might appreciate. Includes interviews with Alan Guth, Roger Penrose and loads of other interesting people. Definitely not pop-sci and fairly up to date. Episode 1 is a bit low quality with unnecessary subtitles, but it gets better as it...
  19. T

    Taylor series representation for $$ \frac{x}{(1+4x)^2}$$

    Homework Statement Find a power series that represents $$ \frac{x}{(1+4x)^2}$$ Homework Equations $$ \sum c_n (x-a)^n $$ The Attempt at a Solution $$ \frac{x}{(1+4x)^2} = x* \frac{1}{(1+4x)^2} $$ since \frac{1}{1+4x}=\frac{d}{dx}\frac{1}{(1+4x)^2} $$ x*\frac{d}{dx}\frac{1}{(1+4x)^2}...
  20. T

    Calculating Coefficients of Fourier Series Homework

    Homework Statement I'm calculating the coefficients for the Fourier series and I got to part where I can't simplify an any further but I know I have to. a_n = \frac{1}{2π}\Big[\frac{cos(n-1)π}{n-1}-\frac{cos(n+1)π}{n+1}-\frac{1}{n-1}+\frac{1}{n+1}\Big]Homework EquationsThe Attempt at a...
  21. T

    B Integral test and its conclusion

    I'm really confused about this test. Suppose we let f(n)=an and f(x) follows all the conditions. When you take the integral of f(x) and gives you some value. What are you supposed to conclude from this value?
  22. P

    Generalised Fourier Series

    Homework Statement By applying the Gram–Schmidt procedure to the list of monomials 1, x, x2, ..., show that the first three elements of an orthonormal basis for the space L2 (−∞, ∞) with weight function ##w(x) = \frac{1}{\sqrt{\pi}} e^{-x^2} ## are ##e_0(x)=1## , ##e_1(x)= 2x## ,##e_2(x)=...
  23. T

    MHB Finding the value at which the series converges

    I need to use the maclaurin series to find where this series converges: $$\sum_{n = 0}^{\infty} (-1)^n \frac{\pi^{2n}}{(2n)!}$$ But I'm not sure how to do this.
  24. T

    MHB Finding the function of a maclaurin series

    I need to find the function for this Maclaurin series $$1 - \frac{5^3x^3}{3!} + \frac{5^5x^5}{5!} - \frac{5^7x^7}{7!} ...$$ I can derive this sigma: $$1 + \sum_{n = 2}^{\infty} \frac{(-1)^{n - 1} 5^{2n - 1} x^{2n - 1}}{(2n - 1)!}$$ But I'm not sure how to get this function from this series.
  25. T

    MHB How do I find the MacLaurin series for $\frac{1}{1 - 2x}$?

    I need to find the maclaurin series of the function $$\frac{1}{1 - 2x}$$. I know $\frac{1}{1 - x}$ is $1 + x + x^2 + x^3 ...$ but how can I use this to solve the problem? I don't think I can just plug in $2x$ can I?
  26. physiclawsrule

    MHB Question about MacLaurin series

    Aren't the Maclaurin series an expansion of a function about 0 f(x) = f(0) + (f '(0) / 1!) * x + (f ''(0) / 2!) * x^2 + (f '''(0) / 3!) * x^3 + ...
  27. T

    MHB Finding a maclaurin series for a function with 'e'

    I need to find the Maclaurin series for $$f(x) = e^{x - 2}$$ I know that the maclaurin series for $f(x) = e^x$ is $$\sum_{n = 0}^{\infty} \frac{x^n}{n!}$$ If I substitute in $x - 2$ for x, I would get $$\sum_{n = 0}^{\infty} \frac{(x - 2)^n}{n!}$$ However, this is wrong, according to the...
  28. T

    MHB Finding Maclaurin series of a natural log function

    I need to find the Maclaurin series of this function: $$f(x) = ln(1 - x^2)$$ I know that $ln(1 + x)$ equals $$\sum_{n = 1}^{\infty}\frac{(-1)^{n - 1} x^n}{n}$$ Or, $x - \frac{x^2}{2} + \frac{x^3}{3} ...$ If I swap in $-x^2$ for x, I get: $$-x^2 + \frac{x^4}{2} - \frac{x^5}{3} +...
  29. T

    MHB Maclaurin series for natural log function

    I'm examining the Maclaurin series for $f(x) = ln(x + 1)$. It is fairly straightforward but there are a few details I'm not getting. So: $$ ln(x + 1) = \int_{}^{} \frac{1}{1 + x}\,dx$$ which equals: $A + x - \frac{x^2}{2}$ etc. or $A + \sum_{n = 1}^{\infty}(-1)^{n - 1}\frac{x^n}{n}$ I'm...
  30. T

    MHB Differentiating a power series

    I need to prove that for $-1 < x < 1$ $$\frac{1}{(1 - x)^2} = 1 + 2x + 3x^2 + 4x^3 ...$$ So, according to the textbook, the geometric series has a radius of convergence $R = 1$ (I'm not sure how this is true). In any case we can compare it to: $$\frac{1}{1 - x} =\sum_{n = 0}^{\infty} x^n$$...
  31. T

    MHB Finding Maclaurin series of a function

    I need to find the Maclaurin series for this function: $$f(x) = (1 - x)^{- \frac{1}{2}}$$ And I need to find $f^n(a)$ First, I need the first few derivatives: $$f'(x) ={- \frac{1}{2}} (1 - x)^{- \frac{3}{2}}$$ $$f''(x) ={ \frac{3}{4}} (1 - x)^{- \frac{5}{2}}$$ $$f'''(x) ={- \frac{15}{8}}...
  32. T

    MHB Finding the interval of convergence for a series with lnn

    So I have $$\sum_{n = 2}^{\infty} \frac{1}{nln(n)}$$ I'm trying to apply the limit comparison test, so I can compare it to $b_n$ or $\frac{1}{n}$ and I can let $a_n = \frac{1}{nln(n)}$ Then I get $$\lim_{{n}\to{\infty}} \frac{n}{nln(n)}$$ Or $$\lim_{{n}\to{\infty}} \frac{1}{ln(n)}$$ Which is...
  33. T

    MHB Finding if a series converges

    I have this series $$\sum_{n =0}^{\infty}\frac{(-1)^n {x}^{2n}}{{2}^{n + 1}}$$ I need to find whether it converges or diverges at $\sqrt{2}$ and $-\sqrt{2}$. I'm not quite sure how to approach this. For $\sqrt{2}$ I have $$\sum_{n =0}^{\infty}\frac{(-1)^n {\sqrt{2}}^{2n}}{{2}^{n +...
  34. Divya Shyam Singh

    Is there a tutorial series for LMS imagine.lab amesim 14?

    I needed to learn LMS imagine lab 14 software but I cannot find any tutorial video on youtube which explains everything from the scratch. Please help me out..
  35. M

    Find the power series in x for the general solution of?

    Homework Statement Find the power series in x for the general solution of (1+x^2)y"+6xy'+6y=0. Homework Equations None. The Attempt at a Solution I got up to an+2=-an(n+3)/(n+1) for n=1, 2, 3, 4, 5, 6... a3=-2a1 a4=0 a5=3a1 a6=0 a7=-4a1 a8=0 The answer in the book says y=a0sigma from m=0 to...
  36. peter010

    DC Series motor: current vs load....

    Hiz lets assum we have a load fixed on the roter of a the 'DC series motor' in the attached photo, where: Vt: DC source voltage (constant) Lf: field's inductive resistance (will be neglected) Rf: field's resistant Ra: Armature resistance Ia= Armature current, If: field current M: back emf (Ea)...
  37. A

    I Taylor Series: What Is the Significance of the a?

    i watched a lot of videos and read a lot on how to choose it, but i what i can't find anywhere is, what's the physical significance of the a, if we were to draw the series, how will the choice of a affect it?
  38. S

    Find Taylor Series for 1/x Around x=3

    Homework Statement Find the Taylor Series for f(x)=1/x about a center of 3. Homework EquationsThe Attempt at a Solution f'(x)=-x^-2 f''(x)=2x^-3 f'''(x)=-6x^-4 f''''(x)=24x^-5 ... f^n(x)=-1^n * (x)^-(n+1) * (x-3)^n I'm not sure where I went wrong...
  39. brainphysics

    Engineering Figuring Out a General Solution for RLC Series Circuit

    So, I have been trying to come up with a general solution for dI/dt in an RLC circuit. I have attached the work I have done so far. I don't know where but I am making a mistake and the waveform is not coming out right. Would really appreciate a look over my work to see if I made any obvious errors.
  40. J

    Other Is Greiner's book series good?

    Hello, I have used Greiner's "Quantum Mechanics: An introduction" and found it to be awesome, bridging the ga between undergraduate and graduate courses. So, I am thinking of buying some of Greiner's book to use for my other courses and I wanted to ask you what your opinions about the books in...
  41. chwala

    Does the Series Converge? A Comparison Test Approach

    Homework Statement determine whether the series below converges. ##\sum_{n=1}^\infty 2^n.n+1,√(n^4+4^n.n^3)## Homework EquationsThe Attempt at a Solution
  42. T

    MHB Finding if a series converges or diverges

    I have this series $$\sum_{n = 1}^{\infty} \frac{\ln\left({n + 4}\right)}{{n}^{\frac{5}{2}}}$$ which I need to find whether it converges or diverges. I can use the limit comparison test and set $a_n = \frac{\ln\left({n + 4}\right)}{{n}^{\frac{5}{2}}}$ and $b_n = \frac{1}{{n}^{\frac{5}{2}}}$...
  43. anemone

    MHB Inequality Of The Sum Of A Series

    Prove \frac{10}{\sqrt{11^{11}}}+\frac{11}{\sqrt{12^{12}}}+\cdots+\frac{2015}{\sqrt{2016^{2016}}}\gt \frac{1}{10!}-\frac{1}{2016!}
  44. T

    MHB Solving Series Limit Problem: Find Convergence/Divergence

    I have this limit: $$\sum_{k = 1}^{\infty} {(\frac{e }{3})}^{k}$$ Which method can I use to find if it converges or diverges?
  45. M

    Need help with wiring motors in series and in parallel

    Hello, I'm working on my kids ride on cars. He has two of them. On the first it was 6V and came with one motor on one wheel. I updated it to 2 motors, one on each wheel and 12Volts. I put in a DPDT switch so he could select high and low. On high the motors are run in parallel and on low they...
  46. parshyaa

    Intro Math "The art of problem solving" textbook series

    I am a 12th grade student. I am new to this series and i know that these are great books. i am going to buy 3 books. the basics , introduction to algebra, introduction to geometry is it necessary to buy solution manual. is it ok to buy these three books for the beginners what about concept...
  47. B

    Did I multiply this infinite series correctly?

    Homework Statement Hi, I have to find the RMS value of the inifnite series in the image below. Homework Equations https://en.wikipedia.org/wiki/Cauchy_product Allowed to assume that the time average of sin^2(wt) and cos^2(wt) = 1/2 The Attempt at a Solution So to get the RMS value I think I...
  48. H

    MHB Find the sum of this series:

    Find the sum of this series: $$ \sum_{n=1}^\infty \frac{n}{(n+1)!} $$ I'm really struggling with this one.. Any help will be highly appreciated. Thanks you.
  49. T

    Given nth partial sum of a series, find a of n and sum

    Homework Statement If the nth partial sum of a series ##\sum_{n=1} ^\infty a_{n}## is ##S_{n} = \frac {n-1} {n+1}## Find ##a_{n}## and ##\sum_{n=1}^\infty a_n## Homework Equations ##S_{n} - S_{n-1}= a_{n}## ##\lim_{n \rightarrow +\infty} {S_{n}} = \sum_{n=1}^\infty a_n = S## The Attempt at a...
  50. maxhersch

    I Find the formula to express the infinite series....

    The problem is to find the general term ##a_n## (not the partial sum) of the infinite series with a starting point n=1 $$a_n = \frac {8} {1^2 + 1} + \frac {1} {2^2 + 1} + \frac {8} {3^2 + 1} + \frac {1} {4^2 + 1} + \text {...}$$ The denominator is easy, just ##n^2 + 1## but I can't think of...
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