# 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. ### Use method of difference to find sum of series

My interest is on the (highlighted part in yellow ) of finding the partial fractions- Phew took me time to figure out this out :cool: My approach on the highlighted part; i let ##kr+1) =x ## then, ##\dfrac{1}{(kr+1)(kr-k+1)} = \dfrac{1}{(x)(x-k)}## then...
2. ### I Question on an infinite summation series

Is the infinite series ##\sum_{n=1,3,5,...}^\infty \frac {1} {n^6}## somewhat related to the Riemann zeta function?The attached image suggest the value to be inverse of the co-efficient of the series.Is there any integral representation of the series from where the series can be evaluated?
3. ### 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
4. ### 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...
5. ### 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...
6. ### 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 ##...
7. ### 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...
8. ### 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...
9. ### 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...

11. ### 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...
12. ### 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}##.
13. ### 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...
14. ### POTW Does the Taylor series for arctan converge at x = 1?

Show that $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$$
15. ### 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...
16. ### 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...
17. ### 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...
18. ### 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...
19. ### 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.
20. ### 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##.
21. ### 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...
22. ### Need help with a (apparently) difficult series

This is the series: $$\sum_{n=1}^{+\infty}\sin(n)\sin\left(\frac{1}{n}\right)\left(\cos\left(\frac{1}{\sqrt{n}}\right)-1\right)$$
23. ### 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...
24. ### 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##?
25. ### 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...
26. ### Integrate or divide the input impedance for transmission lines in series?

Here I list my problem in the attachment.

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?
28. ### 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...
29. ### 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...
30. ### 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...
31. ### 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...
32. ### 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
33. ### 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...