What is Maclaurin series: Definition and 155 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).
I tried diffrentiating upto certain higher orders but didn’t find any way.. is there a trick or a transformation involved to make this task less hectic? Pls help
Homework Statement
Obtain Maclaurin Series for:
f(x) = sin(x2)/x
Homework Equations
f(x) = ∑f(n)(c) (x-c)n / n! (for Maclaurin c = 0)
The Attempt at a Solution
I know that sin(x2) = x2 - (x2*3/3! +...
from the final answer I see, that this is just multiplied to 1/x.
This bothers me...
Homework Statement
Let ##F## be an entire function such that ##\exists## positve constants ##c## and ##d## where ##\vert f(z)\vert \leq c+d\vert z\vert^n, \forall z\in \Bbb{C}##.
Is this question incomplete? My complex analysis course is not rigorous at all and this came up on a past final...
Homework Statement
The problem is attached as pic
Homework Equations
∑(ƒ^(n)(a)(x-a)^n)n! (This is the taylor series formula about point x = 3)The Attempt at a Solution
So I realized that we should be looking at either the 30th,31st term of the series to determine the coefficient. After we...
Homework Statement
I've begun going through Boas' Math Methods in the Physical Sciences and am stuck on problem 1.15.25. The problem is to evaluate
## \lim_{x\to \infty } x^n e^{-x} ##
By using the Maclaurin expansion for ##e^{x}##.
Homework Equations
We know the Maclaurin expansion for the...
I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how.
$$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$
I get most of the function, I just can't see where the ##-1## comes from. Could...
Homework Statement
Note - I do not know why there is a .5 after the ampere. I think it is an error and I have asked my lecturer to clarify.
Homework Equations
The Attempt at a Solution
f(t)=sint2 f(0)=sin(0)2=0
f'(t)=2sintcost f'(0)=sin2(0)=0...
I need to find the Maclaurin series for
$$f(x) = x^2e^x$$
I know
$$e^x = \sum_{n = 0}^{\infty} \frac{x^n}{n!}$$
So, why can't I do
$$x^2 e^x =x^2 \sum_{n = 0}^{\infty} \frac{x^n}{n!} = \sum_{n = 0}^{\infty} \frac{x^2 x^n}{n!} $$
I need to find the function for this Maclaurin series
$$1 - \frac{5^3x^3}{3!} + \frac{5^5x^5}{5!} - \frac{5^7x^7}{7!} ...$$
I can derive this sigma:
$$1 + \sum_{n = 2}^{\infty} \frac{(-1)^{n - 1} 5^{2n - 1} x^{2n - 1}}{(2n - 1)!}$$
But I'm not sure how to get this function from this series.
I need to find the maclaurin series of the function
$$\frac{1}{1 - 2x}$$.
I know $\frac{1}{1 - x}$ is $1 + x + x^2 + x^3 ...$ but how can I use this to solve the problem? I don't think I can just plug in $2x$ can I?
I need to find the Maclaurin series for
$$f(x) = e^{x - 2}$$
I know that the maclaurin series for $f(x) = e^x$ is
$$\sum_{n = 0}^{\infty} \frac{x^n}{n!}$$
If I substitute in $x - 2$ for x, I would get
$$\sum_{n = 0}^{\infty} \frac{(x - 2)^n}{n!}$$
However, this is wrong, according to the...
I need to find the Maclaurin series of this function:
$$f(x) = ln(1 - x^2)$$
I know that $ln(1 + x)$ equals
$$\sum_{n = 1}^{\infty}\frac{(-1)^{n - 1} x^n}{n}$$
Or, $x - \frac{x^2}{2} + \frac{x^3}{3} ...$
If I swap in $-x^2$ for x, I get:
$$-x^2 + \frac{x^4}{2} - \frac{x^5}{3} +...
I'm examining the Maclaurin series for $f(x) = ln(x + 1)$.
It is fairly straightforward but there are a few details I'm not getting.
So:
$$ ln(x + 1) = \int_{}^{} \frac{1}{1 + x}\,dx$$
which equals:
$A + x - \frac{x^2}{2}$ etc. or $A + \sum_{n = 1}^{\infty}(-1)^{n - 1}\frac{x^n}{n}$
I'm...
I need to find the Maclaurin series for this function:
$$f(x) = (1 - x)^{- \frac{1}{2}}$$
And I need to find $f^n(a)$
First, I need the first few derivatives:
$$f'(x) ={- \frac{1}{2}} (1 - x)^{- \frac{3}{2}}$$
$$f''(x) ={ \frac{3}{4}} (1 - x)^{- \frac{5}{2}}$$
$$f'''(x) ={- \frac{15}{8}}...
Homework Statement
\lim\limits_{x \to 0} \left(\ln(1+x)\right)^x
Homework Equations
Maclaurin series:
\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{r+1} \frac{x^r}{r} + ...
The Attempt at a Solution
We're considering vanishingly small x, so just taking the first term in the...
Homework Statement
Solve y''+(cosx)y=0 with power series (centered at 0)
Homework Equations
y(x) = Σ anxn
The Attempt at a Solution
I would just like for someone to check my work:
I first computed (cosx)y like this:
(cosx)y = (1-x2/2!+x4/4!+ ...)*(a0+a1x+a2x2 +...)...
Homework Statement
This question has four parts which may follow up from each other so I incuded all the parts. The real problem I'm having is with d
Consider the function f ang g given by f (x)=( e^x+[e^-x])/2 & g (x) =( [e]^x]-[e^-x])/2
a) show f'(x) = g (x) and g'(x) = f (x)
b) find the...
Hello,
I want to prove that the taylor expansion of f(x)={\frac{1}{\sqrt{1-x}}} converges to ƒ for -1<x<1. If I didn't make a mistake the maclaurin series should look like this:
Tf(x;0)=1+\sum_{n=1}^\infty{\frac{(2n)!}{(2^n n!)^2}}x^n
My attempt is to use the lagrange error bound, which is...
Homework Statement
As I've been going through examples in my textbook they are becoming increasingly lengthy to compute and thus I have resorted to using software to complete the task. For example when computing the series for ##\sin{(\ln{(1+x)})}##...
Homework Statement
Find the Maclaurin series of ##\int_{0}^{x} \cos{t^2} \cdot dt ##
Homework Equations
3. The Attempt at a Solution [/B]
I normally have some idea how to go about solving these but for this one I just can't figure out where to start. I tried doing it with ##\int_{0}^{x}...
Homework Statement
Write the Maclaurin series for ##\frac{1}{(1+x)^{1/2}} ## in ##\sum## form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms of n.
Homework Equations
3. The Attempt at a Solution [/B]...
My homework question is about the first law of blackbody radiation. I have to prove an expansion when
for KT≫ℏw.
After some rewriting of the formula i have (ex-1)-1
because KT≫ℏw, x is close to zero, so i think i should use the maclaurin series.
According Wolfram Alpha the series expansion is...
Homework Statement
Find the Maclaurin series of the function https://webwork.wustl.edu/webwork2_files/tmp/equations/87/63afd4b6f3566e2a90aa420dc5d1821.png
c_3 =
c_4 =
c_5 =
c_6 =
c_7 =
Homework Equations
The Attempt at a Solution
(8x^2)[(9x) - (9x)^3/3! + (9x)^5/5! - (9x)^7/7! + ...]
I got...
Homework Statement
To rephrase the question, given a power series representation for a function, like ex , and its MacLaurin Series, when I expand the two there's no difference between the two, but my question is: Is this true for all functions? Or does the Radius of Convergence have to do with...
Hello! So, I'm having a bit of a problem with an exercise in my Calculus book. I'm supposed to find the Maclaurin series representation of
\frac{1+x^3}{1+x^2}
and then express it as a sum. Am I really supposed to differentiate this expression a bunch of times..? That will be very...
find the first 5 nonzero terms in maclaurin series. (might be binomial)
$f(x)=e^{4x} \sqrt{1+x}$my book doesn't explain it properly and my instructor didnt explain it and I am very stuck and there's going to be one similar to this on the test. help!
Hello.
I am stuck on this question. I'd appreciate if anyone could help me on how to do this.
The question:
Expand the following into maclaurin series and find its radius of convergence.
$\frac{2-z}{(1-z)^2}$
I know that we can use geometric series as geometric series is generally...
Homework Statement
Problem is attached in this post.
Homework Equations
Problem is attached in this post.
The Attempt at a Solution
I came up with the function (1+x)^1/n and tried to derive a maclaurin series out o fit but to no avail, I can't determine what maclaurin series to...
Homework Statement
Problem is attached in this post.
Homework Equations
Problem is attached in this post.
The Attempt at a Solution
I used the Maclaurin Series for sin (x) and got the following series:
π/10 - π^3/6,000 + ... etc.
I can't find a way to simplify the series...
Homework Statement
In attached image.
2. The attempt at a solution
Now, after looking at the solution, the only real conclusion I can come up with is that a Maclaurin series must have x's with non-negative integer value as the exponents, correct? This is because for the the general...
Homework Statement
Use a known Maclaurin series to compute the Maclaurin series for the function: f(x) = x/(1-4(x^2))Homework Equations
1/(1-x) = ∑x^nThe Attempt at a Solution
I tried removing x from the numerator for: x ∑ 1/(1-4(x^2)), which would end up through substitution as x ∑...
Homework Statement
the maclaurin series for f(x) is given by 1/2! - x2/4! + x4/6! - x6/8! + ... + (-1)nx2n/(2n+2)! + ...
a) Let g'(x) = 1-x2 * f(x)
Write the Maclaurin series for g'(x), showing the first three nonzero terms and the general term.
b) write g'(x) in terms of a familiar...
Homework Statement
for this, my coefficient of x^4 which is 8/4! = 1/3 .. but the ans should be 13/24... can you tell me which part contain mistake?
https://i.imgur.com/05NnrdM.jpg
https://i.imgur.com/28Q9o51.jpg
Homework Equations
The Attempt at a Solution
Homework Statement
for this question, i found that my coefficient of x^4 is wrong... after applying the maclaurin series formula, i would get the coefficient of X^4 is -5/96... but the exact ans is -1/96... can anyone check which part is wrong?
Homework Equations
The Attempt at a...
Hey Guys!
I'm stick on this question,
I know that the summation of n=0 to infinity for x^n/n! equals e^x
In the question it wants me to come up with a corresponding summation for the function x^2(e^(3x^2) - 1) … I don't know how to manipulate it to get the -1. I know i can substitute x for...
Homework Statement
Given that ##f(x)=(1+x) ln (1+x)##.
(a) Find the fifth derivative of f(x),
(b) Hence, show that the series expansion of f(x) is given by
##x+\frac{x^{2}}{2} -\frac{x^{3}}{6} + \frac{x^{4}}{12} - \frac{x^{5}}{20}##
(c) Find, in terms of r, an expression for the rth term...
Hi I'm studying for an upcoming exam and I have to find the Maclaurin series for f(x)= ((1-x^2)/(1+x^2))
And I got to admit i feel stuck.
I know i need to find the terms f(0) +f'(0) +f''(0)/2 etc.
Frist of all I can't find the first derivative f´(x) because my TI89 calculater comes up...
Work out the first five derivatives of the function f(x)=sec(x), and hence deduce the Maclaurin series of g(x)=sec(x)(1+tan(x)) up to and including the term of order x^4.
(Hint: why have you been asked for five derivatives of f(x)?)
The Maclaurin series for function g(x) is given by...
The Maclaurin series expansion for ##(1+z)^\alpha## is as follows:
$$(1+z)^\alpha = 1 + \sum_{n=0}^\infty \binom{\alpha}{n}z^n$$ with $$|z|<1$$
What I don't understand is why is ##|z|<1##?
Homework Statement
Determine the first three terms in the Maclaurin series for:
(x2+4)-1
Homework Equations
f(x)=f(a)+f'(a)(x-a)+f''(a)\frac{(x-a)^{2}}{2!}+f'''(a)\frac{(x-a)^{3}}{3!}
The Attempt at a Solution
So I start out with getting my primes of f(x)...
Homework Statement
Find the Maclaurin series for (tanx)2
Homework Equations
f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^{2}+...
The Attempt at a Solution
I don't see how it's reasonable to do this problem without using a computer.
The derivative of (tanx)2 is 2tanxsec2x, then the...
Homework Statement
Use x=-1/2 in the MacLaurin series for e^x to approximate 1/sqrt(e) to four decimal places.Homework Equations
The Attempt at a Solution
\sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + x^2/2 + x^3/6 + ...
For this particular power series, I have:
\sum_{n=0}^\infty...
since the maclaurin series for sin x is alternating in sign (EQ1) so when you square it to get sin^{2}(x) (EQ2) the (-1)^{n} should become (-1)^{2n} (EQ3) which can be simplified down to (EQ4), but when i checked that series at wolframalpha the series was still alternating like: Why is that? So...
Homework Statement
f(x) =ln (1-x^3) / (x^2)
Homework Equations
Using the maclaurin series ln (1 +x) = Ʃ (-1)^(n-1) (x^n)/(n)
The Attempt at a Solution
the maclaurin series for the function i get is [(-1)^(2n-1) (x)^(n)] / (n)
however, the answer according to my prof is...
Here is the question:
Here is a link to the question:
Find the Maclaurin series? - Yahoo! Answers
I have posted a link there to this topic so the OP can find my response.
Homework Statement
find the Maclaurin series and find the interval on which the expansion is valid.
f(x) = ln(1-x2 )
Homework Equations
The Attempt at a Solution
I'm pretty confident in my skill at problems like these, except for this one I am getting an answer different from...
Edit: Never mind. Got it.
Homework Statement
f(x)=\frac { x }{ { (2-x) }^{ 2 } }
Homework Equations
The Attempt at a Solution
I tried finding the first derivative, the second derivative, and so on, but it just keeps getting more complicated, so I suspect I have to use binomial series.
The...