2
votes
1answer
30 views

Evaluating $\ln(\cos x))$ using Taylor expansion

Evaluate $\ln(\cos x)$ at $x_0=0$ and with the order of $n=4$. Noticing that $\ln(\cos x) = \ln(1+ \cos x - 1)$ we can use $\ln(1+x)$ Taylor series. Now, I've read I should use: $$\ln(1+x) = x - ...
1
vote
1answer
30 views

Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
0
votes
0answers
26 views

Product of two $2$-variables Taylor series

Using the standard multi-index notation, suppose we have the two Taylor series $$ f(\theta) := \sum_{|\alpha|=0}^{\infty} a_{\alpha} \theta^{\alpha} $$ and $$ g(\theta) := \sum_{|\alpha|=0}^{\infty} ...
1
vote
1answer
37 views

What can be said about an infinitely differentiable function whose Taylor series diverges?

What can be said in general of an infinitely smooth function whose Taylor series diverges? According to Borel Theorem it is possible to construct such but what kind of property have those special ...
1
vote
0answers
22 views

taylor series expansion, derivatives not continuous

As a part of an excercise I am supposed to find the Taylor series expansion for $(1-t)^{\frac{1}{2}}$ on $[0,1]$. According to the remainder theorem: ...
1
vote
0answers
82 views

A function equal to its Taylor series on an interval is also equal to its Taylor series on a subinterval with different center

Suppse the power series $ \sum_{n=0}^\infty a_n (x-a )^n$ has positive radius of convergence $R$ and thus defines a real analytic fuction $f$ on $(a-R,a+R).$If $x_0$ is a point with $|x_0-a|<R,$ ...
0
votes
2answers
42 views

Convergence of general Taylor series

For any $a,h \in \mathbb{R}$, how can we see the series, $\sum_{k=0}^{\infty} \dfrac{f^{(k)}(a)}{k!} h^k$ converges to $f(a+h)$?
0
votes
0answers
13 views

Lagrange's form of the remainder vs Cauchy's form

So far (while practicing exercises) I've used Lagrange's form of the remainder. Is there a situation when Cauchy's form comes in handy while Lagrange's form fails for some reason? Is there a rule of ...
1
vote
0answers
50 views

Given a function $f$, find the largest $n$ such that $f(x)/x^n$ can be defined at $x=0$ to become differentiable there

Let $f(x) = \ln\left(\frac{x^2}{2}+1\right)+\cos x -1$. Find the largest $n\in\Bbb{N}$ such that there is $C\in\Bbb{R}$ such that: $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C ...
3
votes
3answers
60 views

Show $\forall x>0:\ln(1+x) > x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$

I'm reading a proof which aim to show that: $$\forall x>0:\ln(1+x) > x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$$ the Taylor expansion of $\ln(1+x)$ is (not by chance): $$x - ...
1
vote
2answers
37 views

Multplication of series

My textbook is taking about the Cauchy product and I don't quite understand it and it says that when multiplying series, the sum of the third one is equal to the product of the sums of first two ...
1
vote
1answer
49 views

Taylor series and Lagrange's remainder f(x)=$e^x$

In my textbook the Lagrange's remainder which is associated with the Taylor's formula is defined as: $R_{n}(x)= \frac{(x-a)^n}{n!} f^{(n)}(a + \vartheta (x-a))$, for some $\vartheta$ $\in$ <0 ,1> ...
1
vote
0answers
32 views

PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial ...
1
vote
1answer
29 views

Estimating the remainder of Taylor series written in Lagrange form

Given the function $$f(x) = \ln\left(\frac{1+x}{1-x}\right)$$ Show that the error $f(1/3) - T_n(1/3)$ is at most $55/7776$ My attempt Remainder Term = $[f^{(5)}(x) = 24/(1+x)^5 + ...
5
votes
1answer
128 views

Multiple differentiability from Taylor expansion

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a real function, and let $0\leq n\leq+\infty$. We make the following assumption: For every $a \in\mathbb{R}$ and for $k=n$ (resp., in the case $n=+\infty$: ...
1
vote
0answers
61 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?
2
votes
1answer
88 views

Can we use taylor series to solve difficult equations (example with cos(x)=x)?

Well I saw that the curve of the taylor function series of $\cos(x)$ at $x=0$ marry (it's a french expression to say that is very very near) the curve of $\cos(x)$ between $x=0$ and $x=pi/2$ So if I ...
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 ...
1
vote
1answer
33 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 ...
2
votes
1answer
54 views

Convergence of Taylor series of $\sqrt{1-x}$

Concerning $$\sqrt{1-x} = \sum_{k=0}^{\infty} \left[\prod_{j=1}^k \left(\frac{j-1-\frac{1}{2}}{j}\right)\right]x^k$$ the Taylor series about $x=0$. For $|x|< 1$ this series converges uniformly. ...
1
vote
2answers
78 views

Taylor series of the function $f(x) = (1+x) ^{\frac{1}{x}}$

Good night!! I got this problem and I'd like to find all the mistakes in my statement. This is my prblem. Find the binomial coefficients of $f(x) = (1+x)^{\alpha}$, with $\alpha \in \mathbb{R}$, and ...
2
votes
0answers
27 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
0answers
24 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
2
votes
1answer
25 views

Approximating error using Taylors theorem

I have used a Maclaurin series for the function $f(x) = \cos(2x)$ and have successfully produced: $\dfrac{2^n cos(\frac{n\pi}2)x^n}{n!}$ Now I want to estimate the error in approximating $\cos(2x)$ ...
1
vote
2answers
46 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 ...
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
51 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
224 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 ...
0
votes
3answers
63 views

When does the remainder term in the taylor series go to zero?

When does the remainder term in the taylor series go to zero? Theorem: Let $f\in C^{N+1}([\alpha,\beta])$ and $x,x_0\in(\alpha,\beta)$. Then ...
1
vote
4answers
78 views

If $\displaystyle \sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n= 0$ for any $n$ [closed]

Suppose that $f(x)=\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}$ for all $x$ with the radius of convergence $R>0$. If $\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n=0$ for any $n$.
0
votes
2answers
56 views

Taylor series of $\sin(x)$ converges uniformly on $[-\pi,\pi]$?

According to my notes, the Taylor series of $\sin(x)$ converges uniformly on $[-\pi,\pi]$. I know that the remainder term needs to converge uniformly to $0$ for this to be the case. But I really ...
0
votes
0answers
44 views

Understanding Taylor's Theorem

In our real analysis course, our lecturer has given us the following theorem, which I don't quite understand. It's been given in an odd way, not similar to anything I've found in books or on the ...
2
votes
2answers
52 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
19 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 $Df$ ...
1
vote
1answer
47 views

Where to stop a taylor expansion of a function of more than one variable?

I tried to taylor expand a function of three variables $f(x,y,z)$ around $y=0,z=0$ but I don't find a way to decide where to stop the expansion. Usually when one expands a function of one variable ...
2
votes
1answer
55 views

Show that $|f'(x)| \le \frac{2M_0}{h} +\frac{hM_2}{2}$ and $M_1 \le 2\sqrt{M_0M_2}$

Let $f$ be a twice derivable function and $M_i =\sup_{x \in \mathbb{R}} |f^{(i)}(x)|$ and $|M_0|, |M_2|<\infty $. Preferably using the Taylor series on the interval $[x,x+h]$ show the following ...
0
votes
1answer
65 views

Question about infinitely many times differentiable function.

Could you please give me some hint how to solve this problem: Suppose $f(x)$ is infinitely many times differentiable function on R, $f(0)=f'(0)=f''(0)=0$. Prove : for all $A>0$ exists some ...
2
votes
0answers
18 views

Using derivatives at 0 to define an inner product

Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty ...
0
votes
0answers
66 views

Taylor theorem remainder term

I'm having trouble applying the formula for the remainder in the Taylor's theorem. From Wikipedia we know that for $f(x)=f(a)+f'(a)(x-a)+…\frac{f^{(n)}(a)}{n!}(x-a)^{n}+R$ the remainder $R$ in the ...
1
vote
2answers
45 views

Taylor series of a division-by-zero equation

I need to calculate taylor series of $(\frac1{t^3}+\frac3{t^2})^{1/3} - \sqrt{(\frac1{t^2}-\frac2{t})}$ at $t = 0$ to calculate limit $(\frac1{t^3}+\frac3{t^2})^{1/3} - ...
1
vote
1answer
43 views

Error Estimation Using Taylor's Theorem

I missed the lecture on this and was wondering if someone could explain the steps involved with this problem. I think that what I have to do is evaluate the polynomial up to the second derivative ...
0
votes
1answer
25 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 ...
2
votes
2answers
39 views

Proof with Lagrange Remainder Theorem

I'm trying to prove that $$1+ \frac{x}{3} - \frac{x^2}{9}<(1+x)^{1/3}<1+\frac{x}{3}$$ if $x>0$. Using the Lagrange remainder theorem, I have that $$(1+x)^{1/3}= 1+ \frac{x}{3} - ...
0
votes
0answers
33 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 ...
0
votes
1answer
30 views

A question about Maclaurin polynomial

Could you please give me some hint how to find 3-th degree Maclaurin polynomial of f(x) given f(0)=1 and for all $0<x<\lambda$ $f'(x)=1+f(x)^{10}$. If $\lim_{x\to0}f(x)=f(0)=1$ then $\lim_{x\to ...
0
votes
0answers
15 views

Differentiable function made up of arbitrary points.

Hi all, for this question , my attempt so far is; The function $F$ here is considered as a function of $t$ alone; the value of $x$ is regarded as a constant. Of course, if we change the value of $x$ ...
1
vote
1answer
39 views

Proof concerning logs and taylor series

Prove that if $n$ is a positive integer and $|x| \leq \dfrac{1}{2}n$ then $(i)\quad n\log\left(1+\dfrac{x}{n}\right)=x+Q_{n}(x)$ where $(ii)\quad |Q_{n}(x)|\leq\dfrac{|x|^{2}}{n}$ and deduce ...
1
vote
2answers
54 views

Solving $\lim_{x\rightarrow0}\frac{\log(1+\sin{x})-\log(1+x)}{x-\tan{x}}$ (doubts with Landau notation)

I'm trying to solve the following limit: $$\lim_{x\rightarrow0}\frac{\log(1+\sin{x})-\log(1+x)}{x-\tan{x}}$$ It is pretty straightforward by substituting those expressions by their Taylors ...
3
votes
2answers
112 views

Approximating the cosine by Taylor polynomial

Let $f:=\cos(x)$ I'm asked to find for which values of $x$ we can be sure the 4th degree Taylor polynomial will give an error lesser than $\frac{1}{1000}$. Now, ...