Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

learn more… | top users | synonyms (1)

2
votes
2answers
58 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 ...
0
votes
1answer
23 views

Taylor expansion with non integer exponents in the rest

Consider the function: $$f(x)=\sqrt[3]{8x^2+4x+1}$$ 1) Find $a,b,\alpha,\beta$ such that: $$f(x)=ax^\alpha+bx^\beta+o(x^{-1/3})$$ 2) Find $A=f([0,+∞[)$ and prove that $f:[0,+∞[\rightarrow A$ is ...
2
votes
3answers
101 views

Taylor Theorem inequality

Prove that for all $f\in C^2([0,1])$ with $f(0)=f(1)=0$ and $|f''(x)| \le 1$ $$|f(x)| \le \frac{1}{2}x(1-x)$$ $\forall x \in [0,1]$.
2
votes
2answers
83 views

Taylor Series looks like exponential

guys! I need a little help. I have this series $$ \sum_{k \ge 0} \frac{\Gamma(j)}{\Gamma(j+k/2)}(-t)^k $$ where $j \in \mathbb{N}$. I need to know if the limit of this function when $t$ goes to ...
0
votes
0answers
14 views

generalizing taylor expansions to incorporate arbitrary constraints

Taylor expansions give us a concrete method for approximating functions up to any desired accuracy. But what if I want the resulting function to be constrained somehow? For example, Let $f$ be a ...
2
votes
1answer
32 views

taylor series expansion about undefined point

Question is to find coefficient of $(z-\pi)^2$ in taylor series expansion about $\pi$ of $$f(z)=\frac{\sin z}{z-\pi},z\neq\pi$$ $$=-1,z=\pi$$ now, coefficient of $(z-\pi)^2$ should be ...
1
vote
0answers
16 views

Taylor Series on two variables simultaneously

I need to find Taylor series of u($x_i+h,t_n+k)$ where taylor series of $u(x_i,t_n+k)$ and $u(x_i+h,t_n)$ are given as such. Is it basically the summation of both the terms except $u(x_i,t_n)$? ...
2
votes
2answers
43 views

Application of Taylor's Formula

If we are given that $f''(x) = f(x)$, how do we show that there exist constants $a$ and $b$ such that $f(x) = ae^x + be^{-x}$ for all $x$? A hint is given: We can define another function $g$ by $g(x) ...
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
1answer
22 views

Approximation to the square root

I was reading an article that approximated a square root operator as follows $\sqrt{1+x+y} \cong \sqrt{1+x} + \frac{1}{2}y + O(xy,y^2) $ At first glance that looks like a Taylor series expansion, ...
3
votes
0answers
63 views

Inverse of a power series

I want to find the inverse function of the power series, $$ f(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n+1)!} $$ The only think I can think of that could possibly help is that $$ ...
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: $$ ...
0
votes
2answers
30 views

Doubt about Taylor's polynomial to approximate f(x)

Use the degree 2 Taylor polynomial of $f(x) =$ $\sqrt[3]{1728 + x}$ to approximate $\sqrt[3]{1731}$ and give a bound for the error. To obtain the degree 2 Taylor's polynomial, I computed the second ...
0
votes
1answer
67 views

Taylor Approximation

I have got this question Suppose that $f$ is twice differentiable at every $x\in\mathbb{R}$ and that for every $x\in\mathbb{R}$ $$f''(x) + f(x) = 0.$$ Show that if $f(0) = 0, f'(0) = 0, $ and ...
2
votes
0answers
36 views

What is the 2nd order taylor polynomial of f(x,y)?

I'm just computing the 2nd order taylor polynomial for $f(x,y) = tan(x + 3y + \frac{\pi}{4})$ centered at (3,-1) and wondering if I have done this correctly or if anyone has any suggestions on how I ...
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
35 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 ...
1
vote
3answers
24 views

Question on a Taylor Polynomial

We are asked to generate the taylor polynomial $P(x)$ for $$ f(x) = \frac{e^{{(x-1)}^2}-1}{(x-1)^{2}} $$ about $x=1$ Using substitution into the known taylor polynomial of $e^{x}$ and further ...
0
votes
2answers
35 views

Taylor polynom, residual for symmetric values

When creating the taylor polynom for a $C^3$-function around a certain point i get the formula $f(z+h)=f(h)+hf'(z)+\frac{h^2}{2}f''(z) + \frac{h^3}{6}f'''(z) + R$ Now lets say I create the polynom ...
1
vote
3answers
188 views

Find complicated Taylor Series

According to some software, the power series of the expression, $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$ around $x=0$ is $$\sqrt{x}-x^{3/2}+\mathcal{O}(x^{5/2}).$$ When I try to do it I find that I ...
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
1answer
71 views

Finding an asymptotic expansion for a transcedental equation

I am new around here and was hoping you will be able to help me with the following. I have the equation: $x^3 - 3x^2 +(3-\epsilon ) x + \epsilon = sin(\frac{\pi}{2} x +\frac{\pi \epsilon}{2} ) $ and ...
2
votes
6answers
248 views

Definition of matrix exponential

Is there an alternative definition of a matrix exponential so I can use it to prove that $$e^{A}=\sum_{m=0}^{\infty} \frac{1}{m!}(A)^m \;?$$ Thanks a lot in advance!
3
votes
5answers
101 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 ...
0
votes
1answer
16 views

Taylor Expansion and Log transformation (Time Series)

From: Time Series Analysis with Applications in R by Jonathan D. Cryer and Kung-Sik Chan. Here is the Taylor expansion: $\log Y_t = \sum_{n = 1}^{\infty} (-1)^{n+1} \frac{(Y_t - 1)^n}{n} $. How ...
1
vote
2answers
34 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,
0
votes
1answer
38 views

Dot product of taylor series $\sqrt{1+x}$

I have to prove that $$ \sum_{k=1}^n \alpha_k \cdot \alpha_{n-k+1} = 0, $$ where $n>2$ and $\alpha_k$ is the k-th member in taylor series of $\sqrt{1+x}$. Namely, $$ \alpha_k = ...
0
votes
1answer
35 views

Problem with Taylor (asymptotic) expansion of hyperbolic functions at infinity

(Note: I chose a general title, because I believe this discussion will be applicable to all other hyperbolic functions having an asymptote at infinity, but I will specifically be focusing on ...
1
vote
1answer
58 views

Taylor Series Expansion for $\tan x$

I'm trying to determine the Taylor series expansion for $\tan x$: I know that the $n$th derivative of the expansion must be the same as the $n$th derivative of the function. Please help, I have no ...
5
votes
1answer
177 views

Limits of the solutions to $x\sin x = 1$

Let $x_n$ be the sequence of increasing solutions to $x\sin{x} = 1$. Define $$a = \lim_{n \to \infty} n(x_{2n+1} - 2\pi n) $$ and $$b = \lim_{n \to \infty} n^3 \left( x_{2n+1} - 2\pi n - \frac{a}{n} ...
0
votes
1answer
13 views

Showing a function is identically zero using taylors theorem

Suppose we have a function $x(t)$ that is analytic everywhere, $x(t)\geq 0$, and $x(0)=0$. Suppose further that we know $x(t)$ is locally zero. ie. $x(t)=0$ for some small $t>0$. Is there a way ...
1
vote
1answer
40 views

Proving $\frac{\cos(t \arctan(\sqrt{x}))}{(1+x)^{t/2}}= \sum_{k \ge 0}\frac{\Gamma(t+2k)\Gamma(k+1)}{\Gamma(t)\Gamma(2k+1)}\frac{(-x)^k}{k!}$

$$\frac{\cos(t \arctan(\sqrt{x}))}{(1+x)^{t/2}}= \sum_{k \ge 0}\frac{\Gamma(t+2k)\Gamma(k+1)}{\Gamma(t)\Gamma(2k+1)}\frac{(-x)^k}{k!}$$ This is particularly nice, and apparently can be proved with the ...
2
votes
1answer
87 views

How would I integrate $e^{e^x}$?

Is there a way to integrate: $e^{e^x}$ without using a Taylor or McLaurin Series expansion?
0
votes
0answers
12 views

How can I show that $u=e^{\sigma\sqrt{\Delta t}}$ in the binomial option pricing model

Given that $e^{r\Delta t}(u+d)-ud-e^{2r\Delta t} = \sigma^2\Delta t$ I would like to show that $u=e^{\sigma\sqrt{\Delta t}}$ I know I must somehow use Taylor's approximation $e^x = 1 + x + ...
0
votes
1answer
38 views

Taylor series of a rational function

I am facing some complicated integral, which part of it is $$\frac{z^{M-1}}{(1+(\eta z)^n)^p}$$ I think if I find the taylor series of this part the integral might be solved. So, can someone help me ...
3
votes
1answer
33 views

Finding $\frac{\partial ^8 f}{\partial x^4\partial y^4}$

Given the function $f(x,y)=\frac{1}{1-xy}$ find the value of$\frac{\partial ^8 f}{\partial x^4\partial y^4}(0,0)$. First I developed the function into a taylor series using geometric series ...
0
votes
1answer
85 views

Taylor Expansion of $ \frac{1}{x} $ about x = 0

I'm confused with the following problem: The expression $ \frac{1}{x} $ is clearly not defined at x = 0. However, I read that it could be expressed as a series using the idea $ ...
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 ...
1
vote
1answer
48 views

Does Taylor's theorem apply here?

Let $U\subset \mathbb{R}^n$ be open and $f:U\to \mathbb{R}^n$ with $x\in U$ and $\xi$ sufficiently small. Suppose that the following hold: $f(x+\xi)=\sum_{\alpha=0}^k ...
0
votes
0answers
28 views

Taylor series remainder approximation

Can I approximate the remainder with $(n+\delta)^k$ if $k\not \in \mathbb N$? I think that the answer is yes, at least for points such that their distance from the origin is less than 1, but I can't ...
0
votes
3answers
134 views

Find a power series representation for the function $f(x)=\frac{(x-1)^2}{(3-x)^2}$

I tried to separate it and found the sum of $$\frac{1}{(1-x/3)^2}$$ but then I got stuck with having to multiply my sum with $(x-1)^2$ . I tried looking online but there's close to nothing about ...
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?
0
votes
1answer
33 views

Prove inequality using taylor series

Let $0\leq p \leq 1$ and $\phi(t)=t\log \frac{t}{p} + (1-t)\log \frac{1-t}{1-p}$. Prove $\phi(t)\geq 2(t-p)^2$ for $t\in[0,1]$. Here's how I started. $$\phi'(t) = -\log ...
0
votes
0answers
29 views

Taylor’s theorem with the Lagrange form of the remainder Expansion

Write down the Taylor expansions of $f(0) and f(2)$ using Taylor Theorem with the Lagrange Form of the remainder. Here is the formula. $f(a+h)=f(a)+hf'(a)+ (1/2) h^2f''(a+θh)$ My confusion is ...
0
votes
0answers
54 views

Use Taylor Theorem Special Form to Prove

Suppose $f: \mathbb{R} \rightarrow \mathbb{R} $ is such that both $f'$ and $f''$ exist for all $x \in \mathbb{R}$, Suppose that on [0,2] the inequalities $|f(x)|\leq 1$ and $|f''(x)|\leq1$ hold. ...
1
vote
0answers
62 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 ...
9
votes
7answers
760 views

How are the Taylor Series derived?

I know the Taylor Series are infinite sums that represent some functions like $\sin(x)$. But it has always made me wonder how they were derived? How is something like ...
2
votes
1answer
34 views

Alternating series error bound

The taylor series for $ln(x)$, centered at $x=1$, is $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{(x-1)^n}{n} $$ Let $f$ be the function given by the sum of the first three nonzero terms of this series. The ...
1
vote
0answers
32 views

Finding f'(0) in a taylor series

While doing questions involving taylor series, I accidentally chanced upon an unorthodox, if more difficult way of solving for $f^{(n)}(0)$ of a given taylor series. I am wondering why it works, if ...