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).
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...
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?
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...
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...
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 ##...
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...
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...
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...
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...
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}##.
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...
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...
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...
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...
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...
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.
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...
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...
##\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##?
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...
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?
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...
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...
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...
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...
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...
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}...
##\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...
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...
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.
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)##?
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...
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...
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...
(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...
##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 <...
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
$$ \dfrac{d}{dx} \left((1-x^2) \dfrac{d}{dx}S_{lm} \right) + \left(c^2x^2 + A_{lm} -...
First I got ##f(0)=0##,
Then I got ##f'(x)(0)=\frac{\cos x(2+\cosh x)-\sin x\sinh x}{(2+\cosh x)^2}=1/3##
But when I tried to got ##f''(x)## and ##f'''(x)##, I felt that's terrible, If there's some easy way to get the anwser?
My first attempt was ##... + n^{2} + (n+1)^{2} > \frac {1}{3} n^{3} + (n+1)^{2}##
then we must show that ##\frac {1}{3} n^{3} + (n+1)^{2} > \frac {1}{3} (n+1)^{3}##
We evaluate both sides and see that the LHS is indeed bigger than RHS. However, this solution is inconsistent so I am asking for...
I do not have any reasonable attempts at this problem, as I am trying to figure out how one can get the correct answer when we are not given any values. Maybe if some of you sees a mistake here, that implies that the values from the previous example should be used...
##a_3 = a_1 \cdot k{2}##...
I have tried inserting 0.955 in the above formula for the sum of a geometric series and setting it equal to 3,000,000 (S_n) with n =3. This did not work out well
My second attempt was, considering that the payment is paid every year in the future, to use the convergence formula. There k = 0.955...
I am trying to understand the series expansion of $$(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 ...