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. 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...
  2. 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...
  3. J

    How Do You Find a Recurrence Relation for a Power Series Solution of an ODE?

    I believe I am doing everything right up until the point where I have to try and find a recurrence relation. I honestly have no idea what to do from there. I've listed my work in getting the powers of n and the indicies to all match. Any help appreciated. Here is the original DE...
  4. S

    MHB I hope to sum up this series symbolically

    Dear Colleagues I posted the same post in the group of Analysis. Perhaps it should have been posted here. It is a finite series for which I am seeking the sum. I tried using MATHEMATICA which did not work. Someone told me MAPLE will do it. So if one has it, I shall be thankful. All you have to...
  5. S

    Maple Summation of a Finite Series: Seeking the Sum with MAPLE or Other Software

    Dear Colleagues I hope this post belongs here in calculus. It concerns a finite series for which I am seeking the sum. I tried using MATHEMATICA which didn't accept it. Perhaps if someone has Maple or any other software who can do it. Here it is attached. I shall be most grateful
  6. F

    Link between Z-transform and Taylor series expansion

    Hello, I am reading a course on signal processing involving the Z-transform, and I just read something that leaves me confused. Let ##F(z)## be the given Z-transform of a numerical function ##f[n]## (discrete amplitudes, discrete variable), which has a positive semi-finite support and finite...
  7. chwala

    Find the sum of the series ##\sum_{r=n+1}^{2n} u_r##

    Find question and solution here Part (i) is clear to me as they made use of, $$\sum_{r=n+1}^{2n} u_r=\sum_{r=1}^{2n} u_r-\sum_{r=1}^{n} u_r$$ to later give us the required working to solution... ... ##4n^2(4n+3)-n^2(2n+3)=16n^3+12n^2-2n^3-3n^2=14n^3+9n^2## as indicated. My question is on...
  8. gremory

    A Power series in quantum mechanics

    Just earlier today i was practicing solving some ODEs with the power series method and when i did it to the infinite square well i noticed that my final answer for ##\psi(x)## wouldn't give me the quantised energies. My solution was $$\psi(x) = \sum^{\infty}_{n=0} k^{2n}(\cos(x) + \sin(x))$$...
  9. docnet

    Sum of Infinite Series: Finding the Value of S

    Using the given rule for the ##x_n##, write $$ \sum_l y_l = x_1 + \frac{1}{2} x_2 + \frac{1}{6} x_2 + \frac{1}{3} x_3 + \frac{1}{12} x_2 + \frac{1}{12} x_3 + \frac{1}{20} x_2 + \cdots + \frac{1}{n} x_n $$ $$ = x_1 + \sum_{n=2}^\infty \frac{1}{n(n-1)} x_2 + \sum_{n=3}^\infty \frac{2}{n(n-1)} x_3...
  10. S

    Visualizing Series Bandpass Filters: A Graphical Approach

    this is the circuit this is the theoretical graph
  11. Margarita0076

    Engineering MIPS - Fibonacci Series: Main and fib function

    Write and test the fib function in two linked files (Fib.asm, fib_main.asm). Your solution must be made up of a function called fib(N, &array) to store the first N elements of the Fibonacci sequence into an array in memory. The value N is passed in $a0, and the address of the array is passed in...
  12. Jehannum

    I Exploring the Relationship Between a/b and Geometric Series

    While working on a probability problem I accidentally found this relationship: $$\frac a b = \frac a {(b-1)} - \frac a {{(b-1)}^2} + \frac a {{(b-1)}^3} - \frac a {{(b-1)}^4} + ~...$$ I have done a bit of work on it myself, and have tried to research similar series. It seems to lead to some...
  13. S

    MHB Infinite series involving 'x' has a constant value

    How to prove that \[ \sum_{i=1}^{\infty}\frac{1}{2^{3i}}\left(\csc^{2}\left(\frac{\pi x}{2^{i}}\right)+1\right)\sec^{2}\left(\frac{\pi x}{2^{i}}\right)\sin^{2}\left(\pi x\right)=1 \] for all \( x\in\mathbb{R} \). Using graph, we can see that the value of this series is 1 for all values of x...
  14. Vividly

    B Understanding about Sequences and Series

    Homework Statement:: Tell me if a sequence or series diverges or converges Relevant Equations:: Geometric series, Telescoping series, Sequences. If I have a sequence equation can I tell if it converges or diverges by taking its limit or plugging in numbers to see what it goes too? Also if I...
  15. nmfowlkes

    Potential Difference over Capacitors in Series

    V1 = Q/30 In parallel: C = C1 + C2 V2 = Q/(60 + 90) V1/V2 = 3 V1 = 3V2
  16. H

    Finding 2 solutions of this Bessel's function using a power series

    I have to find 2 solutions of this Bessel's function using a power series. ##x^2 d^2y/dx^2 + x dy/dx+ (x^2 -9/4)y = 0## I'm using Frobenius method. What I did so far I put the function in the standard form and we have a singularity at x=0. Then using ##y(x) = (x-x_0)^p \sum(a_n)(x-x_0)^n##...
  17. chwala

    Finding the Limit of a Series: Exploring Different Patterns and Approaches

    Consider the series below; From my own calculations, i noted that this series can also be written as ##S_n##=##\dfrac {3}{8}##⋅##\dfrac {4^n}{3^n}##. If indeed that is the case then how do we find the limit of my series to realize the required solution of ##1## as indicated on the textbook? I...
  18. A

    Question about the convergence of a series

    Greetings! I have a question about one assumption regarding this question even though I agree with the answer but I have a doubt about A, because when we study the convergence of a serie we use the assymptotic approximation, so why A is not correct? thank you! when we
  19. A

    Finding the Taylor series of a function

    Greetings! Here is the solution that I understand very well I reach a point I think the Professor has mad a mistake , which I need to confirm after putting x-1=t we found: But in this line I think there is error of factorization because we still need and (-1)^(n+1) over 3^n Thank you...
  20. A

    The radius of convergence of a series

    Greetings! I have a problem with the solution of that exercice I don´t agree with it because if i choose to factorise with 6^n instead of 2^n will get 5/6 instead thank you!
  21. mechorigin

    Engineering Having issues series resistive and reactive impedance matching

    The only thing I could think of was using Q=sqrt(6000/50)-1 = 10.9, which then gets me XL=545 and XC=550, or 96.4nH, and 0.32pF at resonance of 900Mhz. I tried seeing if Zin=Zout equation would bring me close, so I tried Z=545+(50-550)=45, then XL for 38.5nH was 217.7, and XC for 2.4pF was 74...
  22. chwala

    Show that the series ##\sum_{n=2}^\infty\dfrac {1}{n^2\ln (n)}##converges

    This is the question, its long since i studied convergence...I need to attempt all the questions (attached)i will therefore make an attempt to answer one part at a time i.e ##a## first. wawawawawa! does not look nice...let me look at my old notes on this chapter then i will respond...
  23. Lorena_Santoro

    MHB Find the missing number in the series please

  24. L

    Multiplication of Taylor and Laurent series

    First series \frac{1}{2}\sum^{\infty}_{n=0}\frac{(-1)^n}{n+1}(\frac{1}{p^2})^{n+1}= \frac{1}{2}(\frac{1}{p^2}-\frac{1}{2p^4}+\frac{1}{3p^6}-\frac{1}{4p^8}+...) whereas second one is...
  25. P

    How to know which is bigger? (Comparing two infinite series)

    Summary:: How to know which one is bigger when n goes to infinity? $$ \sum_{n=1}^\infty \frac {1} {\sqrt {n}(\sqrt {n+1}+\sqrt {n-1})} $$ And: $$ \sum_{n=1}^\infty \frac {1} {\sqrt {n}(\sqrt {n}+\sqrt {n})} $$ I thought at first that the second one is bigger, although, I came to realize, to my...
  26. docnet

    Find the Laurent Series of a function

    (a) i tried to decompose the fracion as a sum of fractions of form ##\frac{1}{1-g}## $$f=\frac{-z}{(1+z)(2-z)}=\frac{a}{1+z}+\frac{b}{2-z}$$ $$a=\frac{1}{3}, b=-\frac{2}{3}$$ $$f=\frac{1}{6}\frac{1}{1+z}-\frac{1}{3}\frac{1}{1-\frac{z}{2}}$$ $$f=\frac{1}{6}\sum_{n=0}^\infty...
  27. K

    I Reversing a series of polarizers

    If light at a known polarity goes through a beam splitting polarizer and then goes through the reverse orientation of that polarizer it will exit with the same polarization that it entered with. What happens if you put light through a series of beam splitting polarizers and then through the...
  28. S

    Determining if a series converges

    The following is my attempt at the solution. Here, I used limit comparison test to arrive at the answer that the series converges. However, the answer sheet reads that the series diverges. I am confused because I cannot figure where my work went wrong… can anyone tell me how the series...
  29. S

    Using comparison tests and limit comparison test

    The answer sheet states that the series converges by limit comparison test (the second way). In the case of this particular problem, would it be also okay to use the comparison test, as shown above? (The first way) Thank you!
  30. F

    MHB How can I develop ln(x) into a series for x >= 1 in fluid dynamics?

    I need to develop $\mathrm{ln}(x)$ into series, where $x \geq 1$, and I don`t know how? In literature I only found series of $\mathrm{ln}(x)$, where: 1. $|x-1| \leq 1 \land x \neq 0$, $ \,\,\,\,\, \mathrm{ln}(x) = x - 1 - \dfrac{(x-1)^2}{2} + ...$ 2. $|x| \leq 1 \land x \neq -1$, $ \,\,\,\,\...
  31. K

    B Conductors in the triboelectric series

    I teach electricity in grade 9. For the concept of conductors, they are described in the textbook as atoms where the outer electrons can easily move from one atom to another (e.g. copper). But I noticed that on the triboelectric series, copper and other metals are listed as having a strong(er)...
  32. R

    Fourier series, periodic function for a string free at each end

    From the statement above, since the ring is massless, there's no force acting vertically on the rings. Thus, the slope is null. ##\frac{\partial y(0,0)}{\partial x} = \frac{\partial y(L,0)}{\partial x} = 0## ##\frac{\partial y(0,0)}{\partial x} = A\frac{2 \pi}{L}cos(\frac{2 \pi 0}{L}) =...
  33. S

    Determining if series converges or diverges

    Is it valid to use limit comparison test to compute the following series? If it is, would my reasoning be valid? Thank you!
  34. Astronuc

    Isaac Asimov's Foundation Series on Apple TV+

    Previously, there was a possibility that HBO would make a series, https://www.physicsforums.com/threads/hbo-will-make-asimovs-foundation.781302/ but, https://www.bbc.com/culture/article/20210920-foundation-the-unfilmable-sci-fi-epic-now-on-our-screens Filmmaker David S Goyer was working...
  35. BerriesAndCream

    Calculating nth Term of Sequences: What Now?

    I don't understand what the question is asking. the nth term of the first sequence i can calculate to be -2n+4, while 2n-24 is the nth term for the second sequence. now what? The question isn't clear.
  36. Andrew1235

    Conservation of energy for a series of elastic collisions

    The speed of the block after the nth collision is $$ V_n=(2e)^n*v_0 $$ By conservation of energy the block travels a distance $$V_n^2/(2ug)$$ on the nth bounce. So the total distance is $$ d=1/(2ug)∗(v_0^2+(2ev_0)^2...) $$ $$ d=1/(2ug)∗(v_0^2/(1−4e^2)) $$ $$ d=1/(2ug)∗(v_0^2∗M^2/(M^2−4m^2))...
  37. N

    Calculate the individual voltage, current and power in series and para

    To determine the voltage I did voltage/number of globes: 24/16 = 1.5V per globe - Not sure if this is correct or not To determine current, I figured out using resistance formulas that the resistance for each set of 8 globes is 15 ohms R = V/I 24/3.2 = 7.5 ohms total resistance 7.5-1 = 2 * 15-1...
  38. L

    Plugging Power Strips Together in Series?

    I have read one shouldn't plug a power strip into another power strip. Why might that be? I don't mean to exceed the number of outlets of the first strip. I just don't have enough space between outlets to plug everything in. Like a 1/2" between the outlets would be fine.
  39. R

    Show that the sum of the finite limits of these two series is also finite

    In the homework I am asked to proof this, the hint says that I can use the triangle inequality. I was thinking that if both series go to a real number, a real number is just any number on the real number line, but how do I go from there,
  40. M

    I Fit a non-linear function to this time series

    I have an experimantally obtained time series: n_test(t) with about 5500 data points. Now I assume that this n_test(t) should follow the following equation: n(t) = n_max - (n_max - n_start)*exp(-t/tau). How can I find the values for n_start, n_max and tau so as to find the best fit to the...
  41. V

    Capacitors in parallel or series

    To me it seems like the formula applies to capacitors of any shape or size, since textbooks never mention any limitations on capacitor type when stating these formulae.
  42. A

    The convergence of a numerical series

    Greetings here is the exercice My solution was as n^2+n+1/(n+1) tends asymptotically to n then the entire stuffs inside the sinus function tends to npi which make it asymptotically equal to sin(npi) which is equal to 0 and consequently the sequence is Absolutely convergent Here is the...
  43. jackiepollock

    Generalisation of terms in a series

    Hello. I'm not sure how the generalisation comes about (where I circle). I assume that r means the the rth derivative of f(x). If that's the case, as I plug 3 = r into this generalisation, the third derivative term should equal to (-1)^3x^7 /7!, but the third derivative term is -1x^3/3...
  44. U

    I Prove series identity (Alternating reciprocal factorial sum)

    This alternating series indentity with ascending and descending reciprocal factorials has me stumped. \frac{1}{k! \, n!} + \frac{-1}{(k+1)! \, (n-1)!} + \frac{1}{(k+2)! \, (n-2)!} \cdots \frac{(-1)^n}{(k+n)! \, (0)!} = \frac{1}{(k-1)! \, n! \, (k+n)} Or more compactly, \sum_{r=0}^{n} (...
  45. bagasme

    Lingusitics Learning Target Language with Mother Tongue-subtitled Series

    Hi, Supposed that Raul was learning Korean, because he would like to work with BTS and BLACKPINK's agency, which requires fluency in Korean. And let's assumed that his mother tongue language is Indonesian. After several months of courses and he adequately understood Korean, he decided to watch...
  46. W

    Series Convergence: What Can the Nth Term Test Tell Us?

    I'm not sure which test is the best to use, so I just start with a divergence test ##\lim_{n \to \infty} \frac {n+3}{\sqrt{5n^2+1}}## The +3 and +1 are negligible ##\lim_{n \to \infty} \frac {n}{\sqrt{5n^2}}## So now I have ##\infty / \infty##. So it's not conclusive. Trying ratio test...
  47. W

    Understanding the Ratio Test for Series and Its Applications

    So I am having some difficulty expressing this series explicitly. I just tried finding some terms ##b_{0} = 5## I am assuming I am allowed to use that for ##b_{1}## for the series, even if the series begins at ##n=1##? With that assumption, I have ##b_{1} = -\frac {5}{4}## ##b_{2} = -...
  48. S

    Bounds of the remainder of a Taylor series

    I have found the Taylor series up to 4th derivative: $$f(x)=\frac{1}{2}-\frac{1}{4}(x-1)+\frac{1}{8}(x-1)^2-\frac{1}{16}(x-1)^3+\frac{1}{32}(x-1)^4$$ Using Taylor Inequality: ##a=1, d=2## and ##f^{4} (x)=\frac{24}{(1+x)^5}## I need to find M that satisfies ##|f^4 (x)| \leq M## From ##|x-1|...
  49. 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
  50. 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...
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