2
votes
2answers
58 views

Differentiating the Taylor expansion of $e^x$

It is well known that a) $\frac{d}{dx}\exp x = \exp x$ and b) $\exp x = \sum\limits_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{3} + ...$. Therefore, it should be possible to ...
0
votes
2answers
37 views

How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
1
vote
0answers
59 views

To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
1
vote
0answers
27 views

please help me completing this proof (Lagrange remainder for Taylor formula)

I'm trying to prove that the remainder of a $n$-th grade Taylor formula is $$R_n=\frac{f^{(n+1)}(\mu) (x-x_0)^{n+1}}{ (n+1)!}$$ where $\mu$ is a value between $x$ and the centre $x_0$. For $n=1$ it ...
3
votes
5answers
105 views

Taylor expansion of $\frac{1}{1+x^{2}}$ at $0.$

I am trying to find the Taylor expansion for the function $$f(x) = \frac{1}{1+x^{2}}$$ at $a=0.$ I have looked up the Taylor expansion and concluded that it would be sufficient to show that ...
2
votes
1answer
48 views

Understanding the proof of Taylor's theorem

I'm trying to understand the proof of Taylor's theorem from here: I already made a question about the remainder part of the theorem and got an answer for it here: Remainder term in Taylor's ...
3
votes
1answer
56 views

Remainder term in Taylor's theorem

I'm trying to understand the remainder in Taylor's theorem from this source: https://proofwiki.org/wiki/Taylor%27s_Theorem/One_Variable/Integral_Version I don't understand the very last parts of ...
1
vote
1answer
29 views

Aftermath of Cauchy's mean value theorem

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
0
votes
1answer
14 views

Question about Peano form of the remainder

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
-1
votes
1answer
34 views

How do you represent f(x+h) and f(x−h) as a Taylor series using the taylor series formula?

I know the answers are below, however i am not quite sure what to substitute as the "a" in the Taylor series formula. $f(x+h)=f(x)+f′(x)⋅h+\frac 12f′′(x)\cdot h^2+\cdots+\frac 1{n!}f^{(n)}(x) \cdot ...
1
vote
2answers
180 views

Solution to curious infinite series

How exactly does one find a closed form to: $$ \sum_{i=0}^{\infty}\left[\frac{1}{i!}\left(\frac{e^2 -1}{2} \right)^i \prod_{j=0}^{i}(x-2j) \right]$$ When expanded it takes on the form $$1 + ...
5
votes
5answers
72 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
1
vote
1answer
66 views

How do I construct such a numerical method for solving ODE?

I am asked to expand $x(t+h)$ and $x(t+2h)$ around $t$ up to the rest term of the third order, find $A, B, C \in \mathbb R$ such that $$x'(t)=\frac{Ax(t)+Bx(t+h)+Cx(t+2h)}{h} + O(h^2)$$ and based on ...
3
votes
2answers
48 views

Substituting for Taylor series

So my question is simple: Why is substitution valid? I mean it seems counter-intuitive to me mainly because of the chain rule. For example: The Taylor series of $e^{x^2}$ is simply done by ...
1
vote
0answers
23 views

Taylor polynomial, Peano form of the remainder of f(x) and its asymptote

Could someone help me complete this or check if my reasoning so far is correct? I'm stuck at finding the oblique asymptote: Write the Taylor polynomial and the Peano form of the remainder of $f(x) = ...
1
vote
2answers
38 views

Does $\sum\limits_{n=1}^\infty \left[n\left(f\left(\frac{1}{n}\right)-f\left(-\frac{1}{n}\right)\right)-2f'(0)\right]$ converge?

Let $f\in C^3([-1,1])$ Is the series $\sum\limits_{n=1}^\infty \left[n\left(f\left(\frac{1}{n}\right)-f\left(-\frac{1}{n}\right)\right)-2f'(0)\right]$ convergent? I'm trying to use Taylor's ...
2
votes
0answers
26 views

Entire periodic $f(z)$ with more than 50 % of the derivatives $0$?

Im looking for a real-entire function $f(z)$ such that for any complex $z$ : $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number. $2)$ $f(z)= 0 + a_1 z + a_2 z^2 + a_3 z^3 + ...$ where more than ...
0
votes
2answers
45 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
4
votes
1answer
49 views

Let $f:[-1,1] \to \mathbb{R}$ be differentiable 3 times, prove $\exists M>0 \ , \ s.t \ f(x) \le Mx^2$

Let $f:[-1,1] \to \mathbb{R}$ be differentiable 3 times, let $f(0)=0$ and $f(x) \ge 0 \ \forall x \in [-1,1]$. Prove: $\exists M>0 \ , \ s.t \ f(x) \le Mx^2$. I separated the proofs to ...
2
votes
1answer
23 views

Using Taylor's series in imporper integrals

Is it possible to simplify an improper integral using Taylor's series? How can I prove this procedure is correct? For example, take $$f(\alpha)=\int_0^{\infty} ...
1
vote
1answer
50 views

Maclaurin series of (1+x)^(1/x)

how can i find the Maclaurin series of $f(x)=(1+x)^{1 \over x}$? $f(0)$ is not even defined, or should I define it as $f(0)=e$? I stopped at the first derivative as it gets terribly messy. thank ...
8
votes
1answer
222 views

Real-analytic periodic $f(z)$ that has more than 50 % of the derivatives positive?

Im looking for a real-analytic function $f(z)$ such that for any $z$ $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number and where $z$ is close to , or onto the real line such that $z$ is in the ...
2
votes
1answer
36 views

Taylor series $\ln(2+x)$ centered at $x=2$

Taylor series $\ln(2+x)$ centered at $x=2$. Is the correct result $$y=\ln \left(4\right)+\sum _{n=1}^{∞}\frac{\left(-1\right)^n}{4^{\left(2^{\Large n}\right)}}\cdot \frac{\left(x-2\right)^n}{n!}\ ?$$ ...
1
vote
2answers
30 views

Meaning of interval of convergence when approximating functions

Let's say I have a Taylor series approximation, $p(x)$, of a function $f(x)$ at $a$: $$ p(x)=\sum_{n=0}^\infty{\frac{f^{(n)}(a)}{n!}(x-a)^n} $$ And that this Taylor series has a radius of ...
3
votes
2answers
117 views

Proving $\forall x\in\mathbb R : \dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$ with Cauchy's MVT

Prove for all $x\in\mathbb R$: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ Mclauren expansion: $$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+R_4(x)$$ ...
0
votes
0answers
20 views

solution check for approximating derivative using a Taylor expansion.

I'm wondering if there's in a mistake in either my reasoning or the given solution for the problem and was hoping to have someone double check this for me. The problem states: Let $g(2)=3$ , ...
0
votes
1answer
38 views

Cubic MacLaurin $e^{x^2}$

Find the Cubic MacLurin expansion of e^{x^2}. First, I tried the sub $t=x^2$ and used the regular expansion for $e^t$. But that was wrong. Can I not do non-linear substituions? My calculations: ...
1
vote
4answers
125 views

Two ways to show that $\sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$

Show that: $\large \sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$ on: $0<x<\frac {\pi}2$ I tried to solve it in two ways and got a little stuck: One way is to use Cauchy's MVT, define $f,g$ ...
1
vote
2answers
48 views

compute taylor series about $x=0$ of $\arctan(e^x -1 )$

hello I am having some issue and need a little guidance with this taylor expansion $$f(x)=arctan(e^x -1)$$ the terms i should get are $x+\frac{x^2}{2}-\frac{x^3}{6}-\frac{11 x^4}{24}-\frac{5 ...
1
vote
3answers
68 views

Taylor expansion at $x=0$ of $\ln(1/(1-x))$

Hello I am having some trouble with the taylor expansion of $$f(x)= \ln \frac1{1-x}$$ Would it be correct to treat the inner part as the following geometric series? ...
1
vote
5answers
101 views

Taylor series for $\sinh1$

I am doing taylor series and I want to do it on $\sinh1$. is there a way to make this problem really simple before I begin? note: $\sinh x= \cfrac{e^x - e^{-x}}2$ Any ideas are really helpful ...
2
votes
2answers
50 views

Trying to solve a Taylor series problem

I have a Taylor series problem, well more precisely a Maclaurin series. I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$ Okay here goes: $$f'(x) = 3xe^{x^3} + 6x e^{{2x}^3}$$ ...
1
vote
1answer
18 views

Linear functional vs. map

A few days ago we were briefly discussing Taylor's theorem in higher dimensions in the lecture. Referring to the expression $f(x)=f(a)+Df(a)(x-a)+$higher order the lecturer said that in general $Df$ ...
4
votes
2answers
73 views

Taylor Series Representation of $e^{1-\cos(x)}$

Hello I was wondering how to simplify this Taylor Series $$ e^{1-\cos(x)} =\sum_{k=0}^\infty\frac{(1-\cos(x))}{k!} ^k\ $$ to where I can write out the first couple of terms which are $ ...
1
vote
0answers
14 views

Why more smooth the function the more precise finite difference method?

As the title, Why more smooth the function the better finite difference method? I guess that if the function is smooth we can better approximate with Taylor series, but formally how this helps? ...
1
vote
4answers
67 views

Taylor Series - general 'what' and 'why' questions

I am a little confused with the Taylor Series at the moment, so please forgive me for my very basic questions. If we were to approximate a function, say $cos(x)$, I let $f(x)=cos(x)$ And I have been ...
0
votes
1answer
58 views

Why is the series expansion of $\sin(x)$ at $x=n\pi$ different to the expansion of $\sin(2x)$ at $x=\frac{n\pi}{2}?$

I'm trying to do a series expansion of $\sin(2x)$ about the point $x=\frac{n\pi}2$ where is an integer. I thought that the expansion would be the same as for $\sin(x)$ about the point $x=n\pi$ but ...
1
vote
1answer
43 views

L'Hospital's rule vs Taylor series

One classical application of Taylor expansions is to obtain polynomial equivalents of complicated functions and use them to compute limits. For example, with Landau notations, we have ...
0
votes
1answer
53 views

How do I compute the Taylor Series for $\arctan(x)$?

I've just stumbled upon Taylor Series on Wikipedia and I've been trying to obtain an expansion for $\arctan(x)$, but I can't manage to see a pattern for the $n$th derivative . Can someone come up with ...
0
votes
0answers
27 views

What is the radius of convergence of the derivative of a smooth Taylor series?

On this website I found that the derivative of a Taylor series has the same radius of convergence as the Taylor series itselves. However, there is no reference added, and I seem to be unable to find ...
0
votes
0answers
25 views

Finding a function with a given Taylor expansion

Is there any function $f(x)$ which has the following Taylor series representation? $$ f(x) = \sum_{k=0}^{\infty}{c_{k} (1 + \frac{2x^{2}}{k})^{-k/2}}. $$ for some coefficients ...
2
votes
0answers
23 views

Plank's first law expansion

I'm getting a little stuck on this question. The question is: show that for $KT \gg h\omega$, the first law of Planck: $\displaystyle U =\frac{h\omega}{e^{(h\omega/KT)}-1} \approx KT - ...
0
votes
2answers
47 views

Find the Taylor series generated by f at x=a.

$f(x) = \frac 1 {9 - x}, a = 3$. The answer in the book is $$\sum_{n = 0}^{\infty} \frac{(x - 3)^n}{6^{n + 1}}$$but I'm not sure how to get the above.
0
votes
1answer
23 views

Taylor series remainder question

Let $f(x)=\frac{\sin(x)}{x}$ when $x\neq 0$ and $f(x)=1$ when $x=0$. Starting with the Taylor polynomial of degree $2n+1$ for $\sin(x)$ and the estimate for the remainder term, show that ...
0
votes
0answers
32 views

Taylor Remainder proof for $e^x$

Prove that if $x\leq 0$ then the remainder term $R_{n,0}$ for $e^x$ satisfies $|R_{n,0}|\leq \frac{|x|^{n+1}}{(n+1)!}$. First, $P_{n,0}(x)=1+x+\frac{x^2}{2!}+\cdots+\frac{x^n}{n!}$ with ...
1
vote
1answer
41 views

Taylor series of $e^{(x-1)^2}$ about $x=1$

How would we find the Taylor series of $e^{(x-1)^2}$ about $a=1$? I tried finding the answer using the Taylor series of $e^x$ about $a=1$ which I was able to do correctly. When I substituted ...
2
votes
3answers
77 views

Evaluate the limit with Taylor series

How one can evaluate following limit: $\lim_{x\to\infty} x(\frac{1}{e}-(\frac{x}{x+1})^x)$ ? I've found this exercise in the chapter about Taylor series, but I have no idea how to solve it.
4
votes
2answers
55 views

Approximating the erf function

I was trying to find an approximate solution to the following: $\DeclareMathOperator\erf{erf}$ $$\frac12 \sqrt{\pi} \erf\left (\frac{x-2}{\sqrt{10}}\right) + \frac12 \sqrt{\pi} \erf ...
4
votes
2answers
231 views

Derivation 9.97 in Jaynes' Probability Theory

In page 298 of Jaynes' Probability Theory: the Logic of Science, equation (9.97), Jaynes says: We expect that, if hypothesis $H$ is true, then $n_k$ will be close to $np_k$, in the sense that the ...
-1
votes
1answer
43 views

Find the Taylor Series generated by $\frac1x$ at $x = a$

Can someone help me find the Taylor series for the following equation: $f(x) = \frac1x$ at $a = 10$