0
votes
3answers
65 views

Taylor expansion for a multivariable function

\begin{align} T(x_1,\dots,x_d) &= \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty \cdots \sum_{n_d = 0}^\infty \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + ...
2
votes
2answers
52 views

High Order Derivative Using Maclaurin Series

Use the Maclaurin series to solve the following: $$ \frac{d^6}{dx^6}(x^4e^{x^2}) $$ I got about halfway through the problem before getting stuck. I am not sure how to solve it... Any advice? Also, ...
0
votes
0answers
27 views

Taylor Series Calc 2

I am not sure how to find a series representation for the natural log. If anyone can show me some helpful steps to solve this problem it would be greatly appreciated. What is the Maclaurin series ...
3
votes
2answers
24 views

Maclaurin Series for a natural logarithm

Can anyone please help me with this question? Find the Maclaurin series and the interval of convergence for $f(x) = \ln(1-7x^9)$ I thought the answer was $$\sum_{n=1}^{\infty} (-1)^n ...
0
votes
2answers
30 views

Taylor Series Maclaurin Series Interval Expansion

Hi! I am currently woking on some clack online homework problem. I really have no idea how to approach this problem. If someone could help me solve this question I would greatly appreciate it!
1
vote
3answers
33 views

Expand a sin(x^3) in Maclaurin's series and find a 30th derivative at (0)

I have a big task and problems with it. I have to expand this function in Maclaurin's series. $$cos(x^{3})$$ I tried expand it as $$\sum_{n=0}^\infty (x^n)/n!$$ but it's for $cos(x)$. So i don't know ...
1
vote
0answers
18 views

Expand a function in Maclaurin's series

Please help me expand this function in Maclaurin's series to find an interval of convergence. I tried to turn it into this expression: but it doesn't give me the result. Help me please it's ...
3
votes
2answers
65 views

Decide convergence of the series

Using Taylor expansion decide convergence of the series: $$\sum_{n=1}^{\infty}(e-(1+{{1}\over{n}})^n)^p = \sum_{n=1}^{\infty}a_n$$ I expanded $a_n$ like this $a_n = (e-(1+{{1}\over{n}})^n)^p = ...
2
votes
2answers
61 views

Taylor series of $\sqrt{1+x}$ using sigma notation

I want help in writing Taylor series of $\sqrt{1+x}$ using sigma notation I got till $1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\ldots$ and so on. But I don't know what will come in ...
1
vote
2answers
20 views

Power series of a function about a non zero point

No clue how to ask questions here so here goes nothing! How do I work towards finding the power series of a function centered about a point a not equal to $0$? The specific question I was asked is to ...
0
votes
2answers
46 views

Retrieving $f(x)$ from Taylor Series

I've been trying to extract the $f(x)$ from the following Taylor series: $$ \sum_{k=1}^{\infty} (-1)^{k}\frac{kx^{k+1}}{3^{k}} $$ I moved things around a bit to make it look like this: $$ ...
3
votes
2answers
50 views

Finding terms of a Taylor series where $f(x)$ is a function with a power

I've been stuck with this Taylor series problem for a while now. We have that $$ f(x) = (1 + x^2)^{-2/3} $$ and it's centered at $0$. So what I thought of doing was the $$ \frac{f^{n}(a)(x - ...
1
vote
1answer
36 views

Is this series G(1/n) convergent or divergent given G(x)?

Suppose $G(x)=\int_0^x\sin{\left(e^s-1\right)}ds$ Does the series $\sum_{n=1}^{\infty}G(\frac{1}{n})$ converge or diverge? I'm not sure how to go about solving this; however in our notes it says ...
2
votes
3answers
38 views

standard Taylor series using substitution

Find Taylor series using substitution about $0$ for $f(x)=\frac{125}{(5+4x)^3}$ by writing $\frac{125}{(5+4x)^3}=\frac{1}{(1+\frac{4}{5}x)^3}$? Determine a range of validity for this series.
3
votes
5answers
104 views

Why don't taylor series represent the entire function?

Say, I have a continuos function that is infinitely differentiate on the interval $I$. It can then be written as a taylor series. However, taylor series aren't always completely equal to the function ...
1
vote
2answers
36 views

what is the Maclaurin Series of this function?

Can anyone explain to me how to find the Maclaurin series of: $$f(x)=(x^2+1)e^{\frac{-x^2}{4}}$$ and why does it converge for every x? thanks,
3
votes
4answers
84 views

Taylor Series of $ \frac{1}{1-x^2} $ about x=2

I am trying to form a taylor series of the following: $ \frac{1}{1-x^2} $ about $x=2$ I tried factoring the equation such that it becomes the following: $ \frac{1}{{(1+x)}{(1-x)}} $ I tried to ...
4
votes
1answer
84 views

How to find the series $\sum_{n=1}^{\infty}\frac{[(n-1)!]^2}{(2n)!}(2x)^{2n},$

Find this sum $$\sum_{n=1}^{\infty}\dfrac{[(n-1)!]^2}{(2n)!}(2x)^{2n}.\qquad (-1\le x\le 1)$$ My idea: let $$f(x)=\sum_{n=1}^{\infty}\dfrac{[(n-1)!]^2}{(2n)!}(2x)^{2n},$$ then we have ...
0
votes
3answers
59 views

Why Taylor series does not converge for all x in the domain of the function

Example: $$ f(x)=\frac{1}{1+x} \qquad x\neq-1 $$ $$ f(x)=1-x+x^2-x^3+x^4-x^5+\;... \qquad |x| < 1 $$ Why Taylor series does not converge for all x in the domain of the function?
1
vote
0answers
70 views

Use Taylor Theorem and Taylor Expansion to Prove

Let $g: R\rightarrow R$ be a twice differentiable function satisfying $g(0)=1, g'(0)=0$ and $ g''(x)-g(x)=0$, for all $x$ in R Fix $x$ in R. Show that there exists $M>0$ such that for all natural ...
0
votes
2answers
55 views

Using Lagrange form of the remainder with cosh

I am trying to find "$\cosh 4$ using the sixth partial sum ($n=5$) of the Maclaurin series" for the function. I am also trying to use "the Lagrange form of the remainder to estimate the number of ...
0
votes
2answers
48 views

What is series coefficient for $f(x)=\csc^2 x - \frac1{x^2}$?

What is general formula for Maclauren series expansion for $f(x)=\csc^2 x - \frac1{x^2}$ ?
0
votes
1answer
30 views

Approximation of $\sqrt{1+wi}$

How can $\sqrt{1+wi}$ be approximated? where $-\infty<w<\infty$; My aim here is getting rid of the square root. I've tried binomial, Maclurin and Taylor series around various points. but they ...
0
votes
1answer
35 views

What is the series expansion of $f(z)\cdot\exp\left({s\,\log(z)}\right)$?

For analytic $f$, how can I represent the expression $f(z)\cdot\exp\left({s\,\log(z)}\right)$, i.e. $f(z)\cdot z^s$ in the form $$\sum_{n}^\infty\left(\sum_{k}^\infty a_k s^k\right)z^n,$$ at least ...
3
votes
1answer
55 views

How to show that $e^{x+y} = e^x e^y$ by series expansion [duplicate]

I know that $e^xe^y=e^{x+y}$ but I want to show it by expanding the exponentials in MacLaurin Series. $$ \left(\sum_{n=0}^{\infty} \frac{x^n}{n!}\right) \left(\sum_{m=0}^{\infty} ...
0
votes
2answers
33 views

How can I see $\frac{1}{1 - e^{-u}} = \sum_{k=0}^{\infty} e^{-ku}$?

How can I see $$\frac{1}{1 - e^{-u}} = \sum_{k=0}^{\infty} e^{-ku} ?$$ I know it's related to Taylor series, but I don't get it.
1
vote
0answers
43 views

Error Term of the Taylor series of cosh

I have the Taylor series of cosh $$\sum_{n=0}^\infty \frac {x^{2n}} {(2n)!}$$ and I know that this series converges for all x, but now I want to know if the series represents the function, in other ...
1
vote
1answer
93 views

Taylor series of $\frac 1 {1+x^2}$

I have to construct the Taylor series of $$\frac 1 {1+x^2}$$ around $0$ and $1$ and analyze the convergence in both cases. Also (but this is a consequence of the previous series) I have to construct ...
4
votes
2answers
156 views

Express $(1-z)^{-1}$ as a power series around $z_0=-1+i$.

I need to express $(1-z)^{-1}$ as a power series in powers of $(z+1-i)$. I would like some guidance on the complex analogue of power series and in writing out this particular case. Many thanks for ...
3
votes
1answer
64 views

Showing $|a_k | \le 1$

Let $A$ be the closed unit disk $A= \{z \in \mathbb{C}: |z| \le 1 \}$. Suppose $f$ is an entire function whose Taylor series centered at the origin is $$\sum_{k=0}^{\infty} a_kz^k$$ and that $f$ maps ...
1
vote
2answers
41 views

How to prove a series is greater than zero over an interval?

Show that the series $\sum\limits_{k=0}^\infty \frac{(-1)^k(x^{2k+1})}{(2k + 1)!}$ is greater than zero for $0<x\leq \sqrt{6}$ For a function to show something was greater than zero over an ...
3
votes
0answers
86 views

Loss of Significance problems - Taylor Expansion

(2) This question addresses the notion of loss of signi´Čücance. You are encouraged to revisit the Taylor series expansion that you have learned in calculus, as you will need to apply it here. Explain ...
1
vote
1answer
57 views

Taylor series about (3+x)/(x+4)*exp(-x) expanded at x = - 4. How do i replicate what I see in wolfram?

I'm puzzled by wolfram alpha's results. If i ask 'series $\frac{3+x}{4+x} (\exp (-x))$ expanded at -4' It will return a series. What method does it use to do this? I'm familiar with Taylor ...
1
vote
2answers
112 views

Evaluate the sum of the series

Was given the following infinite sum in class as a question* (while we were talking about taylor series expansions of $\ln(1+x)$ and $\arctan(x)$: $$1 + \frac12 - \frac23 + \frac14 + \frac15 - \frac26 ...
0
votes
3answers
52 views

Maclaurin series for $f(x)=\frac{1}{1+x+x^2} $

What is the Maclaurin expansion of $f(x)=\dfrac{1}{1+x+x^2} $? Thank you! Edit: By multiplying both terms with $ (1-x) $ I got to $\dfrac{1}{1-x^3}-\dfrac{x}{1-x^3}$. Is it correct to transform ...
1
vote
1answer
47 views

Maclaurin Series Complex Numbers

I'm having trouble getting to the right solution on the function ${z^2\over (1+z)^2}$ ${z^2\over (1+z)^2}$ = ${z^2}$${1\over (1+z)^2}$ = ${z^2}$${1\over (1+z)(1+z)}$ = ${z^2}$${A \over (1+z)}$ + ...
0
votes
0answers
66 views

taylor series of $\sin t/t$

I know that the taylor series expansion around zero of $\sin t/t$ is: $\sum_{k=0}^{\infty} (-1)^k \frac{(t^{2k})}{(2k+1)!}$ , I need to find its radius of convergence. I saw a few solutions that claim ...
0
votes
2answers
32 views

Tayors series exansion of $(1 - x )^n$ where $0<x<1$ and $n \ge 0$

I want to find Taylor's series or Maclaurin's series expansion of the following. $$(1 - x)^n \ \text{ where }\ \ 0 < x < 1 \text{ and }\ n \ge 0$$ will it be same as that for $$(1 + x)^n ...
3
votes
1answer
213 views

Proof of the correctness of Taylor series

I am looking at the proof provided on the wiki page for taylor series http://en.wikipedia.org/wiki/Taylor%27s_theorem#Proof_for_Taylor.27s_theorem_in_one_real_variable One of the proof provided is ...
2
votes
1answer
95 views

Convergence radius of $\sqrt{\cos(z)}$

Compute the first 3 non zero terms of the Taylor expansion of $\sqrt{\cos(z)}$ at $z=0$ and determine its convergence radius, considering only the principal branch of the square root. I've computed ...
4
votes
5answers
230 views

Taylor expansion of $e^{\cos x}$

I have to find the 5th order Taylor expansion of $e ^{\cos x}$. I know how to do it by computing the derivatives of the function, but the 5th derivative is about a mile long, so I was wondering if ...
1
vote
3answers
70 views

General form for the series expansion of $e$

I've found a lot of series expansions of the Napier's constant. I was wondering if a general form for this could be devised. They all have a trend and similarities. I've been trying but I've been ...
0
votes
1answer
67 views

calculate interval of convergence

How do I calculate the interval of convergence of $$ \frac{1+x}{1-x} $$ I made it into a taylor series expansion using first principles and the sum is this $$\sum_{n=-\infty}^\infty \left( \{ ...
1
vote
1answer
53 views

Maclaurin Series of $\int_0^x \cos t^2\,dt$

Find the Maclaurin Series for $\int_{0}^{x}\cos t^2\,dt$. $$\cos(x) = \sum\frac{(-1)^n x^{2n}}{2n!}$$ I'm trying this: $$\cos^2 x = \sum\frac{(-1)^n x^{4n}}{(2n!)^2}$$ How would you solve this ...
5
votes
3answers
273 views

Evaluating $\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$

How can we obtain following formula? $$\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}=\frac{1}{3}e^{\frac{-x}{2}}x\left(e^{\frac{3x}{2}}-2\sin\left(\frac{\pi+3\sqrt{3}x}{6}\right)\right).$$ I think if we ...
1
vote
3answers
160 views

First $3$ non-zero terms of the Maclaurin Series $\frac{1}{\sqrt{4+x^3}}$

Since each derivative will be multiplied by $3x^2$, are all the terms of this Maclaurin series $0$?
1
vote
2answers
83 views

question about taylor series

Can someone explain why 1 and 2 use different Taylor series? Why i cant use $1/(1+r)$ = $\sum_{n=0}^{inf}(-1)^n r^n$ on 2,vice versa?
1
vote
1answer
37 views

Maclaurin series of $\ln(2+x^2)$

Find the Maclaurin series of $\ln(2+x^2)$. I know that $\displaystyle\ln(1+x) = \sum_{n=1}^\infty\frac {(-1)^{n-1}} {n} x^n $ So is $\displaystyle\ln(1+x^2) = \sum_{n=1}^\infty \frac ...
2
votes
0answers
103 views

Compute the first five non-zero terms of the Taylor series about $a=4$ for $f(x)=\sqrt{x}$

This is my first Taylor Series problem and I want to make sure I completed it correctly. Here is the question: Compute the first five non-zero terms of the Taylor series about $a=4$ for ...
1
vote
0answers
147 views

Taylor's Formula vs. Taylor's Inequality

In my calculus book, Essential Calculus, and in class we were using Taylor's formula to approximate the remainder in Taylor polynomials but I am having a bit of trouble understanding the intuition ...