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. G

    Second order differential - Tanks in series cooling coil

    I'm stuck on a problem: T1 = dT2/dt + xT2 - y T2 = (Ae^(-4.26t))+(Be^(-1.82t))+39.9 I'm unsure how to proceed
  2. T

    B A conjecture on conjectures

    Any set of a series of numbers consisting of increasing integer members, all of which are determined by a common proposition or characteristic, will always be infinite in size. Examples… Prime numbers Mersenne primes Odd perfect numbers(if they exist) Zeroes of the Zeta function Regardless...
  3. S

    Confused about capacitor discharge

    Consider the above diagram. Once the first capacitor is charged, clearly it will have a voltage ##E##. Then when the switch is flipped, the cell no longer matters (there is no complete circuit which goes through the cell), so we have the first capacitor connected to the second one, and it looks...
  4. RChristenk

    Logarithmic Series question for finding ##\log_e2##

    By definition: ##\log_e(1+x)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}- \cdots ## ##(1)## Replacing ##x## by ##−x##, we have: ##\log_e(1-x)=-x-\dfrac{x^2}{2}-\dfrac{x^3}{3}- \cdots## By subtraction, ##\log_e(\dfrac{1+x}{1-x})=2(x+\dfrac{x^3}{3}+\dfrac{x^5}{5}+ \cdots)## Put ##...
  5. P

    On pointwise convergence of Fourier series

    So, the function is piecewise continuous (and differentiable), with (generalized) one-sided derivatives existing at the points of discontinuity. Hence I conclude from the theorem that the series converges pointwise for all ##t## to the function ##f##. I've double checked with WolframAlpha that...
  6. P

    I How does the ratio test fail and the root test succeed here?

    The series that is given is $$\frac12+\frac13+\left(\frac12\right)^2+\left(\frac13\right)^2+\left(\frac12\right)^3+\left(\frac13\right)^3+\ldots.$$ Now, it's easy to see these are two separate geometric series, however, Spivak claims the ratio test fails because the ratio of successive terms...
  7. L

    Checking series for convergence

    Hi, I am having problems with task d) I now wanted to check the convergence using the quotient test, so ## \lim_{n\to\infty} |\frac{a_{n+1}}{a_n}| < 1## I have now proceeded as follows: ##\frac{a_{n+1}}{a_n}=\frac{\Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2}}{3^{k+1}} \cdot...
  8. justin___

    Finding an open-cicuit voltage, why is resistor in series ignored?

    I found how to get the solution to this question (the answer is 200V), but I don't understand why we ignore the 30kOhm resistor when using analysing the circuit. Because it is in series with the open voltage, wouldn't there be some voltage drop across the resistor that would affect the...
  9. Euge

    POTW A Series Converging to a Lipschitz Function

    Prove that the series $$\sum_{k = 1}^\infty \frac{(-1)^{k-1}}{|x| + k}$$ converges for all ##x\in \mathbb{R}## to a Lipschitz function on ##\mathbb{R}##.
  10. F

    Springs connected to a mass in series?

    I found the answer for the springs in parallel, but not for the ones in series. I believe I don't understand how the forces are interacting properly. Here's a force diagram I drew. Everytime I try to make equations from this though my answer dosen't make sense. The mass m has a gravititoanl...
  11. Euge

    POTW A Series Representation for π

    Show that $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$$
  12. Al-Layth

    Book recommendation for techniques of evaluating Series?

    TL;DR Summary: I am looking for a good thorough book that is devoted to assembling and explaining techniques of evaluating series. evaluating series is a very big problem for me right now. I know nowhere near as much about it as I do integration, and the main reason for this is that its quite...
  13. carlsondesign

    I What is the official name for a Field Series in mathematics/physics?

    I've been working on developing infinitesimal recursion (what I call continuous hierarchy), but I ended up arriving at "field series" instead. My searches didn't seem to come up with anything reasonable (battlefield the video game series), so I'm wondering what the official name for a field...
  14. PhysicsRock

    Why Capacitors in Parallels vs. Series: Coaxial Capacitor Case

    So my idea was to separate the capacitor into two individual ones, one of length ##l - a## filled with a vacuum and one of length ##a## filled with the glass tube. The capacitances then are $$ C_0 = \frac{2 \pi \varepsilon_0 (l-a)}{\displaystyle \ln\left( \frac{r_2}{r_1} \right)} $$ for the...
  15. J

    A Margules' Power Series Formula: Deriving Coefficients

    Margules suggested a power series formula for expressing the activity composition variation of a binary system. lnγ1=α1x2+(1/2)α2x2^2+(1/3)α3x2^3+... lnγ2=β1x1+(1/2)β2x1^2+(1/3)β3x1^3+... Applying the Gibbs-Duhem equation with ignoring coefficients αi's and βi's higher than i=3, we can obtain...
  16. Euge

    POTW A Test for Absolute Convergence of a Series

    Let ##\{a_n\}_{n = 1}^\infty## be a sequence of real numbers such that for some real number ##p > 1##, ##\frac{a_n}{a_{n+1}} = 1 + \frac{p}{n} + b_n## where ##\sum b_n## converges absolutely. Show that ##\sum a_n## also converges absolutely.
  17. Euge

    POTW Fourier Series on the Unit Interval

    Evaluate the Fourier series $$\frac{1}{\pi^2}\sum_{k = 1}^\infty \frac{\cos 2\pi kx}{k^2}$$ for ##0 \le x \le 1##.
  18. S

    I Geometry of series terms of the Riemann Zeta Function

    This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i## The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate...
  19. S

    Need help with a (apparently) difficult series

    This is the series: $$\sum_{n=1}^{+\infty}\sin(n)\sin\left(\frac{1}{n}\right)\left(\cos\left(\frac{1}{\sqrt{n}}\right)-1\right)$$
  20. F

    Looking for a particular function

    TL;DR Summary: I want to find a function with f'>0, f''<0 and takes the values 2, 2^2, 2^3, 2^4,..., 2^n Hello everyone. A professor explained the St. Petersburgh paradox in class and the concept of utility function U used to explain why someone won't play a betting game with an infinite...
  21. M

    A Infinite series of this type converges?

    ##\sum_{n=1}^\infty n^{-a}## converge s for ##a\gt 1## - otherwise diverges. Is there any theory for ##a_n##? For example ##a_n\gt 1## and ##\lim_{n\to \infty} a_n =1##. How about non-convergent with ##\liminf a_n=1##?
  22. A

    A Completeness of the formal power series and valued fields

    I had difficulty showing this no matter what I tried in (a) I am not getting it. Here for p(t) in K[[t]] : ## |p|=e^{-v(p)} ## where v(p) is the minimal index with a non-zero coiefficient. I said that p_i is a cauchy sequence so, for every epsilon>0 there exists a natural N such that for all...
  23. L

    I Taylor Expansion Question about this Series

    Can you please explain this series f(x+\alpha)=\sum^{\infty}_{n=0}\frac{\alpha^n}{n!}\frac{d^nf}{dx^n} I am confused. Around which point is this Taylor series?
  24. chwala

    Solving the Fourier cosine series

    My question is; is showing the highlighted step necessary? given the fact that ##\sin (nπ)=0##? My question is in general i.e when solving such questions do i have to bother with showing the highlighted part... secondly, Can i have ##f(x)## in place of ##x^2##? Generally, on problems to do...
  25. chwala

    Solve the problem involving sum of a series

    Attempt; ##\dfrac{1}{r(r+1)(r+2)} -\dfrac{1}{(r+1)(r+2)(r+3)}=\dfrac{(r+3)-1(r)}{r(r+1)(r+2)(r+3)}=\dfrac{3}{r(r+1)(r+2)(r+3)}## Let ##f(r)=\dfrac{1}{r(r+1)(r+2)}## ##f(r+1)= \dfrac{1}{(r+1)(r+2)(r+3)}## Therefore ##\dfrac{3}{r(r+1)(r+2)(r+3)}## is of the form ##f(r)-f(r+1)## When...
  26. chwala

    Solve the problem involving sum of a series

    My attempt; ##r^2+r-r^2+r=2r## Let ##f(r)=(r-1)r## then it follows that ##f(r+1)=r(r+1)## so that ##2r## is of the form ##f(r+1)-f(r)##. When ##r=1;## ##[2×1]=2-0## ##r=2;## ##[2×2]=6-2## ##r=3;## ##[2×3]=12-6## ##r=4;## ##[2×4]=20-12## ... ##r=n-1##, We shall have...
  27. S

    Switching between parallel and series connections (solar)

    We're off grid at 57degrees north. Our source of electricity is solar panels, with a diesel generator as backup. The solar has served us well, until this November, where we had almost 8 weeks of 0 sunshine. I got sick of running the generator. It's noisy, it needs refueling, smells... So I...
  28. supak111

    Calculate current in a 120 VAC circuit with a series 10uF capacitor

    Hi can someone tell me please how much current is passed though below circuit: 120 AC 60Hz mains power going through a 10uF 500v capacitor in series
  29. tworitdash

    A How to sum an infinite convergent series that has a term from the end

    From my physical problem, I ended up having a sum that looks like the following. S_N(\omega) = \sum_{q = 1}^{N-1} \left(1 - \frac{q}{N}\right) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)} I want to know what is the sum when N \to \infty. Here...
  30. H

    For what values of ##\beta## does the series converge?

    For what values of β the following series converges $$ \sum_{k=1}^{\infty} k^{\beta} \left( \frac{1}{\sqrt k} - \frac{1}{\sqrt {k+1}}\right) $$ I thought of doing it like this $$ \frac{k^{\beta} }{\sqrt k} - \frac{k^{\beta} }{\sqrt{k+1}}$$ $$0 \lt \frac{k^{\beta} }{\sqrt k} - \frac{k^{\beta}...
  31. P

    A Prove a series is periodic

    ##\sum _{n=0}^{\infty }\:\frac{sin\left(2^nx\right)}{2^n}## I have to show the series is periodic, ##2\pi ## ( I think ), it is related to fourier - analyze fourrier course at academics, so I might be right, or just periodic, we learned only periodic of ## 2\pi ## ( of course also continuous...
  32. P

    V Space With Norm $||*||$ - Fourier Series

    Hi, a question regarding something I could not really understand The question is: Let V be a space with Norm $||*||$ Prove if $v_n$ converges to vector $v$. and if $v_n$ converges to vector $w$ so $v=w$ and show it by defintion. The question is simple, the thing I dont understand, what...
  33. P

    Series investigation: divergence/convergence

    Hi everyone! It's about the following task: show the convergence or divergence of the following series (combine estimates and criteria). I am not sure if I have solved the problem correctly. Can you guys help me? Is there anything I need to correct? I look forward to your feedback.
  34. A

    I Can we solve a non-autonomous diffeq via Taylor series?

    I've occasionally seen examples where autonomous ODE are solved via a power series. I'm wondering: can you also find a Taylor series solution for a non-autonomous case, like ##y'(t) = f(t)y(t)##?
  35. Shreya

    Capacitor Network - Series or Parallel?

    My textbook solution states that 1 & 2 are in parallel and so is 3 & 4 and those 2 are in series. That is, (1 P 2) S (3 P 4). My thinking is such: points A & B are of same potential, say V, C & D are of same potential, say x and E & F are are of same potential, say 0. So I can say that 1 and 3...
  36. S

    Sicilian coffee in the "Inspector Montalbano" TV series

    In the "Inspector Montalbano" TV Series, the characters often drink coffee in small cups and it is poured from a small silver colored container. I've read on the internet that typical Sicilian coffee is expresso. Is the series consistent with that? Do homes in Sicily usually have expresso...
  37. H

    Doubt regarding the series ##\sum [\sqrt{n+1} - \sqrt{n}\,]##

    We're given the series ##\sum_{n=1}^{\infty} [ \sqrt{n+1} - \sqrt{n} ]##. ##s_n = \sqrt{n+1} - 1## ##s_n## is, of course, an increasing sequence, and unbounded, given any ##M \gt 0##, we have ##N = M^2 +2M## such that ##n \gt N \implies s_n \gt M##. Thus, the series must be divergent. But...
  38. warhammer

    Question on Emission Spectra/Spectral Series | Atomic Physics

    (I need help with the 2nd part as I can answer the theory part properly). For E=4 eV we can find the wavelength of emitted photon. E= 4 eV = 6.4087e-19 J Using E= hc/λ we get λ=310 nm (approx) My doubt is that this should fall in the Balmer Series but we know that the lowest wavelength value...
  39. H

    Learning to use the Cauchy criterion for infinite series

    ##s_1=2## ##s_2=4## ##s_3=5.333## ##s_4=5.9999## ##(s_n)## is increasing, but unable to guess a bound. Let's see if Cauchy criterion can do something. For n>2, $$ s_{n+k} - s_n = \frac{2^{n+1} }{(n+1)!} + \frac{ 2^{n+2} }{(n+2)!} + \cdots \frac{2^{n+k} }{(n+k)!} $$ $$ s_{n+k} - s_n <...
  40. dim_d00m

    A Recurrence relations for series solution of differential equation

    I am currently looking at section IIA of the following paper: https://arxiv.org/pdf/gr-qc/0511111.pdf. Eq. (2.5) proposes an ansatz to solve the spheroidal wave equation (2.1). This equation is $$ \dfrac{d}{dx} \left((1-x^2) \dfrac{d}{dx}S_{lm} \right) + \left(c^2x^2 + A_{lm} -...
  41. esrever10

    Taylor Series Expansion of f(x) at 0

    First I got ##f(0)=0##, Then I got ##f'(x)(0)=\frac{\cos x(2+\cosh x)-\sin x\sinh x}{(2+\cosh x)^2}=1/3## But when I tried to got ##f''(x)## and ##f'''(x)##, I felt that's terrible, If there's some easy way to get the anwser?
  42. A

    Series inequality induction proof

    My first attempt was ##... + n^{2} + (n+1)^{2} > \frac {1}{3} n^{3} + (n+1)^{2}## then we must show that ##\frac {1}{3} n^{3} + (n+1)^{2} > \frac {1}{3} (n+1)^{3}## We evaluate both sides and see that the LHS is indeed bigger than RHS. However, this solution is inconsistent so I am asking for...
  43. A

    Convergence of a Series: Radius and Endpoints

    Greetings According to my understanding: if x converges in 4 means that the series converges -1<x+3<7 but the solution says C Any hint? thank you!
  44. A

    Converging geometric series

    I do not have any reasonable attempts at this problem, as I am trying to figure out how one can get the correct answer when we are not given any values. Maybe if some of you sees a mistake here, that implies that the values from the previous example should be used... ##a_3 = a_1 \cdot k{2}##...
  45. A

    Trust Fund problem using series and sequences

    I have tried inserting 0.955 in the above formula for the sum of a geometric series and setting it equal to 3,000,000 (S_n) with n =3. This did not work out well My second attempt was, considering that the payment is paid every year in the future, to use the convergence formula. There k = 0.955...
  46. Arman777

    A Series expansion of ##(1-cx)^{1/x}##

    I am trying to understand the series expansion of $$(1-cx)^{1/x}$$ The wolframalpha seems to solve the problem by using taylor series for ## x\rightarrow 0## and Puiseux series for ##x\rightarrow \infty##. Any ideas how can I calculate them ...
  47. T

    A Laurent series for algebraic functions

    Hi, I'm writting because I sort of had an idea that looks that it should work but, I did not find any paper talking about it. I was thinking about approximating something like algebraic functions. That is to say, a function of a complex variable z,(probably multivalued) that obeys something...
  48. S

    I Taming a Divergent Series -- But how does it work?

    A convergent version ( i.e. convergent in the critical strip) of the traditional series for the Riemann Zeta is derived in the video linked at the bottom. It gives the correct numerical values (at least along the critical line, where I tried it out). But although it works numerically, I'm...
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