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 (2)

1
vote
2answers
27 views

Taylor series for multivariable functions

To expend the function of multiple variables $$ f({\bf x})=f(x_1,x_2,\dots,x_n):\mathbb R^n\to\mathbb R $$ in Taylor series around $\bf 0$, we have $$ f({\bf x})=f({\bf 0})+Df({\bf 0})\cdot{\bf ...
1
vote
1answer
17 views

Proofs using the Taylor expansion, zero series and limits

If we have the function $f: (0,\infty) \rightarrow\ R$, $f(x) = e^{\frac{-1}{x^2}} $ Show that for $n≥0$, there exists a polynomial function $p_{(n)}(t)$ such that $f^{(n)} (x) ...
1
vote
1answer
40 views

How to expand the Taylor series of functions of several vectors?

We know that the Taylor series expansion of the function of several scalars around zero is $$ f(x,y)=f(0,0)+f_x(0,0)\cdot x+f_y(0,0)\cdot y+\frac{1}{2!}f_{xx}(0,0)\cdot x^2+\dots $$ Then, how about ...
4
votes
4answers
229 views

Sine of argument with large n approximation

I have worked an integral and reduced the integral to $$\frac{n \pi+\sin\left ( \frac{n \pi}{2} \right )-\sin\left ( \frac{3 \pi n}{2} \right )}{2n \pi}$$ I want to show that for $$n\rightarrow ...
1
vote
0answers
58 views

Taylor Expansion of Power of Cumulative Log Normal Distribution Function - Show Lagrange Remainder tends to Zero

QUESTION I am looking to find a simplification of the expression below. I have attempted this using the Taylor series. The question then remains if we can show the Lagrange remainder goes to zero. I ...
-2
votes
1answer
56 views

Why is this proof for Taylor's Remainder theorem not correct?

I am not exactly sure on how to post math equations in the question box so I have all my following information on a google document: ...
6
votes
1answer
6k views

Prove Taylor expansion with mean value theorem

On http://vapour-trail.blogspot.com/2006/03/brief-explanation-of-taylor-series-via.html one can find an hint at how to derive Taylor expansions from the mean value theorem. The process goes as ...
1
vote
1answer
43 views

Taylor Series Polynomial Proof using Induction

If $f : \mathbb R \to\mathbb R $ is a polynomial function of degree $n$ with $a \in\mathbb R$. Show that the $n$-th Taylor polynomial $P_{f,a,n}$ of $f$ at $a$ is equal to $f$. I know that I need to ...
0
votes
0answers
23 views

Taylor and Laurent equal in analytic domain: $\sum_{n=0}^\infty\frac{F^n(0)}{n!}z^{-n}=\sum_{n=0}^\infty\frac{F^n(0)}{n!}z^{-n}$?

If I have a taylor series around zero that looks like this: $\sum_{n=0}^\infty\frac{F^n(0)}{n!}z^{-n}$ Can I claim that this is equal to the first half of the Laurent series: ...
2
votes
1answer
34 views

Finding $w_1,w_2,b$ such that $w_2 \sigma(w_1 x + b) \approx x$

Let $\sigma(z) = 1/(1+e^{-z})$. How can I find $w_1, w_2, b$ such that $w_2 \sigma(w_1 x + b) \approx x$ for $x \in [0,1]$? The hint provided was to rewrite $x = \frac{1}{2}+\Delta$, assume $w_1$ is ...
2
votes
1answer
44 views

Find $\lim\limits_{x\to 0}(\cos(xe^x)-\ln(1-x)-x)^{1/x^3}$

Find $$\lim\limits_{x\to 0}(\cos(xe^x)-\ln(1-x)-x)^{1/x^3}$$ $$\lim\limits_{x\to 0}(\cos(xe^x)-\ln(1-x)-x)^{1/x^3}=e^{\lim\limits_{x\to 0}\frac{\ln(\cos(xe^x)-\ln(1-x)-x)}{x^3}}$$ Using Taylor ...
5
votes
3answers
190 views

Poisson Process - non-zero probability of more than one arrival

Quoting Bertsekas' Introduction to Probability: An arrival process is called a Poisson process with rate $\lambda$ if it has the following properties: a) Time homogenity - the probability ...
0
votes
1answer
24 views

Meromorphic Function on Extended Plane

How do I prove that every meromorphic function on the extended plane is a rational function?
0
votes
0answers
20 views

Fluid Flow and Uniform Convergence of Taylor Series

I'm reading through a text on fluid flow and Laplace's equation, and it makes a statement that I do not understand and would really like to clarify. Here's the setup: Let $u:A \rightarrow R^2$ a ...
1
vote
2answers
33 views

Derive Taylor Series for $f(x)$

I am learning to derive the Taylor Series for $f(x)$, and I cannot remember how to do the following integral. $\int_{x_0}^x \left(x-x_0\right) \, dx $ to get the following solution ...
0
votes
0answers
33 views

Different forms of remainder in Taylor series

In the literature, it is common to find different forms of the remainder function in a truncated Taylor series. To name a few: Integral form Lagrange form Cauchy form First, can you tell me any ...
4
votes
3answers
129 views

What is the 90th derivative of $\cos(x^5)$ where x = 0?

Trying to figure out how to calculate the 90th derivative of $\cos(x^5)$ evaluated at 0. This is what I tried, but I guess I must have done something wrong or am not understanding something ...
2
votes
2answers
34 views

Using Maclaurin series, finding the value of an infinite sum…

I want to find $$\sum\limits_{n = 1}^\infty {{{{{({1 \over 2})}^n}} \over {n(n + 1)}}} $$ The book that has this problem in says to use the Maclaurin series for $(1 - x)\log (1 - x)$. I don't ...
1
vote
1answer
47 views

Maclaurin's Series for $\sec(x)$ with help of Maclaurin's series for $\tan(x)$

Is there any way to derive Maclaurin's series for $\sec(x)$ with the help of Maclaurin's series for $\tan(x)$? As we know, the Maclaurin's Series for $\tan(x)$ is: ...
0
votes
1answer
22 views

Expansion of $x \log \left(\frac{l+x}{x} \right)$ about x=0

I've read that $$x \log \left(\frac{l+x}{x} \right)=x \log \frac{l}{x} + O(x^2).$$ I tried to derive this using the usual Taylor series method but kept getting a division by zero. Could anyone ...
0
votes
1answer
26 views

Maclurin series for $\sin^2(x)$

I am trying to find the maclurin series expansion for $\sin^{2}x$. First I used the half angle identity: $$\frac{1-\cos(2x)}{2}=\sin^{2}x$$ Then substituted in the maclurin series for $\cos(2x)$ to ...
1
vote
1answer
29 views

Taylor expanding $f(y+\epsilon U1 + {\epsilon}^{2} U2,t,\epsilon)$ in $\epsilon$

How would one Taylor expand $\epsilon f(y+\epsilon U1 + {\epsilon}^{2} U2,t,\epsilon)$ in $\epsilon$? Somehow the professor obtained the first few terms to be: $\epsilon f(y+\epsilon U1 + ...
-3
votes
1answer
28 views

Taylor Series Exercise

What method should I use to find the Taylor series of $f(x)=\frac{x+2}{2-3x}$ with center 2? Here's what I did: Let $y=x-2$ ...
0
votes
1answer
67 views

Finding a function that is harmonic in an annulus,

The problem statement is: Suppose that the real series $∑_0^{∞} a_n$ and $∑_0^{∞} b_n$ converge absolutely. Part 1 Prove that there is a function $u(r,θ)$ which is harmonic in $1<r<2$ and ...
2
votes
1answer
51 views

Real analytic way to explain why the radius of convergence of $1/(1+x^2)$ is small

For any series expansion of $\frac{1}{1+x^2}$, the disc of convergence is blocked by the two singularities on $+i$ and $-i$. A series expansion about $0$ gives a radius of convergence of $1$. Is ...
4
votes
1answer
41 views

How to find the Maclaurin series for $f(x) = \frac{1}{1 + \sin(x)}$?

I have that $\frac{1}{1 + x} = 1 - x + x^2 - x^3 + ...$ So then $\frac{1}{1 + \sin(x)}$ should be $ 1 - \sin(x) + \sin^2(x) - \sin^3(x) + ...$ but clearly this is not the case. So how does ...
1
vote
1answer
37 views

complex series expansion for $f(z)=\frac{1}{z-1}$

Expand the function $f(z)=\frac{1}{z-1}$ as as a series around $z_{0}$ in two regions a) $$|z-z_{0}| < |1-z_{0}|$$ b) $$|z-z_{0}| > |1-z_{0}|$$ and find coefficient $a_{n}$ is each case. I ...
0
votes
2answers
47 views

Power Series Approximation of ln(x)

I am working on building a small embedded calculator, and am working on adding a natural logarithm function that utilizes only + and -. I have worked out the power series representation of ln(x) as ...
0
votes
1answer
26 views

Why can the first term of the Taylor series expansion of $\cos(\theta)$ be written in the way below?

Why can the first term of the Taylor series expansion of $\cos(\theta)$ be written as $\cos(\theta_0) - (\theta - \theta_0) \sin (\theta_0)$?
3
votes
1answer
122 views

Use Taylor expansion to determine the leading error term in a quadrature

I'm working on the last problem in an assignment, and need some guidance on what to actually start by doing. The question is asking me to use taylor expansion to determine the leading error term ...
4
votes
3answers
87 views

Why does the expansion of $e^x$ appear to arise in the formula for derangement of $n$ things $D_{n}=n!\sum_{k=0}^n \frac{(-1)^k}{k!}$

I was recently toying with wolframaplha with the expansion of $e^x$ and I noticed a strange thing that on keeping $x=-1$ (if it is allowed!!!).. I get on the RHS a strange looking infinite expression ...
1
vote
0answers
36 views

Show that $\exists \xi\in(0,1)$ satisfying certain condition

Suppose $f\in C^3[-1,1]$ with $f(-1)=0,f(1)=1$ and $f'(0)=0$. Show that for any $a\in\mathbb{R}$, there eixsts $\xi=\xi(a)\in(0,1)$ only depending on $a$ such that $$f'(\xi)-1=a(f(\xi)-\xi).$$ It ...
4
votes
2answers
3k views

Third order term in Taylor Series

What is the third order term in the Taylor Series Expansion? I know it will just be third order partial derivatives but I want to know how is it expressed in a compact Matrix notation. For instance ...
2
votes
1answer
52 views

Solution of the functional equation $g(x)g(z) = g(x+z)+g(x-z)$

What is the solution for the following functional equation? $g(x)g(z) = g(x+z)+g(x-z)$ The solution given is: $g(z) = 2\cos(z)$. In the derivation of the result (using Taylor expansion), there is ...
0
votes
2answers
26 views

Taylor approximation and composition

I have a general and a specific question about the composition of Taylor series. Let's say we have $f(x)$ and $g(x)$. We know that the normal composition of functions is something like this: $g ...
11
votes
3answers
3k views

Why doesn't a Taylor series converge always?

The Taylor expansion itself can be derived from mean value theorems which themselves are valid over the entire domain of the function. Then why doesn't the Taylor series converge over the entire ...
0
votes
2answers
54 views

If $f \in C^{\infty}$ and $f^{(k)}(0)=0$ for all integers $k \ge 0$, then $f \equiv 0$.

I thought this was true since, $f(x)=f(0)+f'(0)x+f''(0) \frac {x^2}{2!} + \dots$ But I am wrong. Where did I make mistake?
2
votes
2answers
54 views

Could someone check my solution for finding constant of a difference quotient?

So the question was, Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be three times differentiable and $f'''$ is bounded, find constants $a,b,c$ such that $$f''(x) = \lim_{h\rightarrow 0} ...
0
votes
1answer
29 views

Taylor polynomials: Why does $R_{n,x_0}(x) = o(|x-x_0|^n)\implies \lim_{x\rightarrow x_0}R_{n,x_0}= 0$?

Given a remainder term $R_{n,x_0}(x)$ of a n-degree Taylor polynomial at $x_0$ $$f(x) = T_{n,x_0}f(x)+R_{n,x_0}f(x)$$ Why does $$R_{n,x_0}(x) = o(|x-x_0|^n)\implies lim_{x\rightarrow x_0}Rn_{n,x_0}= ...
1
vote
1answer
38 views

Strengthened version of Taylor's theorem?

Let $f$ be a continuous real-valued function on $[a,b]$ that is $n+1$ times differentiable on $(a,b)$ and such that $f^{(1)}, f^{(2)},\ldots,f^{(n+1)}$ are bounded on $(a,b)$ and ...
2
votes
3answers
56 views

Approximation by using Taylor Polynomials - why?

Could anyone tell me why would I want to approximate a function $f$ by using its Taylor expansion (is it the same as saying approximation by Taylor polynomials?), if I have the exact formula of the ...
2
votes
1answer
38 views

Computing the Taylor expansion of the square root of cos(z),

Let $\large f(z)=\sqrt{cosz}$ with the branch of the square root chosen so that $f(0)=1$. Consider the power series expansion of $f(z)$ in powers of $z$. Part 1) Compute the first three non-zero ...
-2
votes
1answer
7k views

How does one approximate $\cos(58^\circ)$ to four decimal places accuracy using Taylor's theorem?

When one needs to compute say $\cos (58^\circ)$ with an error of at most $10^{-4}$, how does one go about it? What is an appropriate centre of the Taylor expansion, and how does one determine the ...
0
votes
0answers
13 views

Bounding an expression

I am trying to figure out an upper bound on the following expression $$(1 + \epsilon)^{\frac{A}{1+\epsilon} - B}$$ where $\epsilon \in (0,1)$, $A \in (0,1)$ and $B \in \{0, 1\}$. I tried doing the ...
1
vote
1answer
53 views

Taylor polynomial approximation - Interval of convergence

Find the Taylor polynomial of order $n$ of the cosine function around $x=0$. Then find the largest interval in which the sequence of polynomials $\{p_n\}$ converges to $f(x) = \cos(x)$. I am ...
1
vote
0answers
21 views

Taylor series Integral

When can we use Taylor series expansion and write $\int_0^{\infty} \log(f(x+\alpha x)) dx = \int_0^{\infty}\log(f(x)+\sum_{n=1}^{\infty}\frac{f^{n}(x) (\alpha x)^n}{n!}) dx$? I think, first the ...
0
votes
1answer
24 views

How to show that $1+ \sum \limits _{n=1} ^\infty \frac {x^n} n$ converges pointwise?

I am having trouble showing that the taylor series for $-\ln(1-x)$ converges pointwise on $[0,1)$. I have that the $k$ derivative is $\dfrac {(k-1)!} {(1-x)^k}$. This gives that the Taylor series ...
0
votes
2answers
54 views

Find Taylor Series of $\frac{1}{1+z^2}$ around $1$

For $f(z)=\dfrac{1}{1+z^2}$ find the Taylor series centered at $1$. While I know I could use partial fractions or perhaps maneuver this problem by adding constants, I would really like to use the ...
2
votes
0answers
26 views

Suppose that $|f(z)| \leq e^{-1/|z|}$ for all $z\neq 0$. Prove that $f=0$.

Suppose f is entire function such that $$|f(z)| \leq e^{-1/|z|}$$ for all $z\neq 0$. Show that $f=0$. Hint: Consider the Taylor series of $f$ about $0$ and recursively show that all coefficients ...