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).
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.
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...
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...
1. The problem:
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Rules:
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Homework Statement
Classify the singularities of
##\frac{1}{z^{1/4}(1+z)}##
Find the Laurent series for
##\frac{1}{z^2-1}## around z=1 and z=-1
Homework EquationsThe Attempt at a Solution
So for the first bit there exists a singularity at ##z=0##, but I'm confused about the order of this...
Do (i), (ii) and (iii) apply to conditionally convergent series as well? I feel like they don't. But the book seems to say that they do because it doesn't "state otherwise".
Homework Statement
Classify the singularities of ##\frac{1}{z^2sinh(z)}## and describe the behaviour as z goes to infinity
Find the Laurent series of the above and find the region of convergence
Homework Equations
N/A
The Attempt at a Solution
I thought these two were essentially the same...
Consider a sequence with the ##n^{th}## term ##u_n##. Let ##S_{2m}## be the sum of the ##2m## terms starting from ##u_N## for some ##N\geq1##.
If ##\lim_{N\rightarrow\infty}S_{2m}=0## for all ##m##, then the series converges. Why?
This is not explained in the following proof:
Homework Statement
From the given ans , i knew that it's conditionally convergent (by using alternating test) i can understand the working to show that it's conditionally convergent . But , i also want to show it as not absolutely convergent ...
Homework EquationsThe Attempt at a Solution...
Homework Statement
Here is my equation that I want to find bs values
Homework EquationsThe Attempt at a Solution
I convert sin to cos.
for bs at s=0 I get
and if s is not zero I can't derive a clear answer for bs.
but in electronic engineer book that I read it wrote
Homework Statement
This is for a differential equations class I'm taking and we're talking about the method of Frobeneus, Euler equations, and power series solutions for non-constant coefficients. The ODE is:
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Write ##sin(3x-x_0)## as its Fourier representation. By doing a suitable integral or otherwise, find the possible values of its Fourier coefficients. You may find the following useful:
##sin(\alpha-\beta) =...
Homework Statement
This is for differential equations with nonconstant coefficients and I wasn't so great at series and sequences in calculus so when I came across this example problem I wasn't sure how they got to their final form. If someone could explain it to me that would be really...
Homework Statement
Cassify the singularities of e^\frac{1}{z} and find the Laurent series
Homework Equations
e^\frac{1}{x} =\sum \frac{(\frac{1}{x})^n}{n!}
The Attempt at a Solution
Theres a singularity at z=0, but I need to find the order of the pole
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I have a random set of time series data that is calculated after applying an algorithm to a main random time serie data, and really need to extract all the possible characteristics from the set. The goal is to measure those characteristics and perform some statistical graphs based on those...
Homework Statement
How are the coefficients of the Fourier series modified for a function with a period 2πT?
Homework Equations
a0 = 1/π ∫π-π f(x) dx
an = 1/π ∫π-π f(x) cos(nx) dx
bn = 1/π ∫π-π f(x) sin(nx) dx
The Attempt at a Solution
I tried letting x= t/T
so dx = dt/T and the limits x = ±...
Find the sum for the series
$$\frac{5}{3}+2+\frac{12}{5}+...$$
This equals
$$\frac{25}{15}+\frac{30}{15}+\frac{36}{15}+...$$
So the numerator increases by 4+k from the previous numerator
But unable to set up
$$\sum_{k+1}^{\infty}f(x)$$
The series should go to $\infty$ since the terms only...
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An increasing sequence that is made of 4 positive numbers, The first three of it are arithmetic series. and the last three are geometric series. The last number minus the first number is equal to 30. Find the sum...
Homework Statement
The following function is periodic between -π and π:
f(x) = |x|
Find the Coefficients of the Fourier series and, by examining the Fourier series at x=π or otherwise, determine:
1 + 1/32 + 1/52 + 1/72 ... = Σ∞j=1 1/(2j - 1)2
Homework Equations
f(x) = a0/2 + ∑∞n=1 ancos(nx) +...
Homework Statement
Here is the problem description:
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Homework Statement
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Homework Statement
The optical power of a HeNe -laser is ##P_0 = 5.0mW## and the wavelength ##\lambda = 633nm##. The emitted light is linearly polarized. As the laser beam travels through two in-series -polarizers, the power detected behind the second polarizer ##P_2 = 1mW## . If the first...
[Note: Thread has been moved to the homework forums by a mentor]
This is the Given problem
This is my solution part 1
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This is my solution part 2
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Homework Statement
I'm trying to calculate the Fourier Series for a periodic signal defined as:
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Homework Equations
Fn = 1/T ∫T f(t)cos(kwοt + θk)[/B]
cn/2 + ∑k=1k=∞(cn)cos(kwοt+θk)
cn= 2|Fn|
θk=∠Fn
The Attempt at a Solution
I got Cn =...
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Homework Statement
for d/dx(e^x) , the series should be start from 2 rather than 1 , right ? because when k = 1 , the circled part would become 0 , the series for (e^x) is 1 + x + ...
Homework EquationsThe Attempt at a Solution
Homework Statement
http://imgur.com/12LbqWL
Part b
Homework EquationsThe Attempt at a Solution
Since it says the first four terms, not nonzero, the first four terms would be 0-(1/3-0)+2/9(x-2)-1/9(x-2)^2
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Homework Statement
I was watching a PatrickJMT video on ratio test and he gave this problem. I solved it before he did, and he got that it was divergent. He didn't simplify it initially, so our methods of approach are different. Did I do something wrong? I checked my calculator to make sure all...
Homework Statement
f(x) = x , 0 <x<1/2
1/2 , 1/2 < x <1
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Hello Everyone,
I will try to explain what am I doing here and I hope someone will understand.
ACF - autocorrelation function
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Mod note: Moved from a technical forum section, so missing the homework template
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$$
\sum_{{\rm n}=0}^\infty \left (-\sqrt x \right )^n \ \ \rm ?$$
Find the interval of convergence?
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