1
vote
2answers
77 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
1answer
43 views

Finding Laurent series where given annulus is not in a singularity

I'm given a problem where I need to calculate the Laurent series of $f(z)$ inside the given annulus $$ f(z) = {1\over z^3(z-1)}; \quad 1 < |z| < 2 $$ From online resources(videos, notes) I ...
0
votes
0answers
10 views

Taylors formula

I have a circle $K(a,\epsilon) \subset \Omega $ and for $ \parallel \Delta x \parallel \lt \epsilon $ we look at $ \Delta f = f(a+ \Delta x) - f(a) $ Now I look at the function $ F:[0,1]\rightarrow ...
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 ...
0
votes
0answers
54 views

Taylor series approximation using for a pdf

I have an question which links Taylor series to expectation and variance, but I'm really not sure what it's asking me to do. X is exponential with rate 1. The question asks me to use a three term ...
2
votes
1answer
51 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
62 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 ...
1
vote
3answers
30 views

Find the first three terms of the taylor expansion of $\frac{cos(z)}{1 + z^2}$

The question is: Find the first three non zero terms for the taylor series for $\frac{\cos(z)}{1 + z^2} $ around $z_0 = 0$ What I've done so far is let $f(z) = \frac{\cos(z)}{1 + z^2}$ Then I let ...
1
vote
3answers
58 views

Taylor development of $\arctan(\cos(x))$ near $0$

How would I find the "Taylor development of $\arctan(cos(x))$ near $0$ at order $5$?" I am translating that from french, so I am not sure how I have to call it it english. By order $5$ I mean that I ...
2
votes
2answers
34 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
2answers
25 views

Maclaurin series for (cosx-1)/(x^2)

The solution for this is -1/x+x^2/4!-x^2/6!......, but I'm not sure how to derive this Maclaurin series from cos x. The solution just divided each term in the Maclaurin series for cos x by x^2, and ...
0
votes
0answers
10 views

Find secondary taylor poly and error value.

Question. (1) Find the second Taylor polynomial $P_{2}(x)$ for the function $f(x)=e^{x}cos x$ about $x_{0}=0$. (a) Use $P_2(0.5)$ to approximate $f(0.5)$. Find an upper bound for error ...
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
1answer
25 views

Expanding functions to Taylor series

I need to expand the following functions to a Taylor series and find the radius, and I'm not sure how to do so: (already solved similar questions, but stuck with those.) $f(z) = {\frac{z-1}{3-z}}$ ...
3
votes
1answer
24 views

Help with Taylor Series

I am trying to find a Taylor series for the following function: ${1\over 1-9x}$ centered at c = 7 I browsed through my Calc II book and found that I can use the general formula for a Taylor series ...
1
vote
2answers
35 views

If $y = e^{x(y-1)}$ then $y \approx 1-2(x-1)$ when $0<x-1<<1$

Assume $y = e^{x(y-1)}$. Then $y \approx 1-2(x-1)$ when $0<x-1<<1$ I thought of something like that: $$ e^{x(y-1)} = e^{-2(x-1)}e^{xy+x-2}=(1-2(x-1)+O(x-1)^2)e^{xy+x-2}$$ But I failed ...
-1
votes
2answers
65 views

proof that Even powers of an odd function's taylor polynomial vanish

Let $f$ be a $k$ times continuously differentiable function defined on a neighborhood of $0 \in \mathbb{R}$. Show that if $f(-x) = -f(x) \forall x \in \mathbb{R}^n$, then the coefficients of the ...
0
votes
0answers
63 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
5answers
225 views

Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$

Prove that $$\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$$ The exercise provides the following hint: $\,\,\displaystyle{n \choose k}={n\choose n-k}$. Any help?
1
vote
0answers
36 views

Question about asympotic expansions!! please help!

Question: Find the constants $$a_0, a_1, a_2$$ in the asympotic expansion $$\int_0^x t\sqrt{ln(t)} dt$$ = $a_0(x^2)(lnx)^\frac 12$ + $a_1\frac {x^2}{(lnx)^\frac 12}$ + $a_2\frac {x^2}{(lnx)^\frac ...
1
vote
0answers
35 views

Find the right degree of the Maclaurin polynomial of $e^x$

Here is my question: What degree Maclaurin polynomial of $e^x$ must be taken to guarantee an estimate of $e$ to within $1 \times 10^{-6}?$ I know that the error term is: ...
1
vote
0answers
39 views

Multivariate Taylor Polynomial

The Exercise: Calculate the Taylor polynomial of degree 3 of $f(x,y,z)=x^5y^4z^3$ at $(1,1,1)$ in an arbitrary direction $h$. Use Taylor's theorem to get a bound on the remainder when using this ...
1
vote
2answers
47 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 ...
2
votes
4answers
98 views

Find the taylor expansion of $\sin(x+1)\sin(x+2)$ at $x_0=-1$, up to order $5$.

Find the taylor expansion of $\sin(x+1)\sin(x+2)$ at $x_0=-1$, up to order $5$. Taylor Series $$f(x)=f(a)+(x-a)f'(a)+\frac{(x-a)^2}{2!}f''(a)+...+\frac{(x-a)^r}{r!}f^{(r)}(a)+...$$ I've got my ...
1
vote
3answers
285 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 $0$?
2
votes
1answer
67 views

Linearize a simple ODE

This is homework. I have $\displaystyle \qquad S\frac{dh(t)}{dt} + \frac{1}{R}\sqrt{h(t)} = q(t)$ and need to linearize it. Setting all derivatives to zero, I get the steady-staty value of $h - ...
1
vote
1answer
39 views

f is a smooth function, and $M_n$ is the sup of $f^{(n)}$. Show if $\lim_{n \to \infty} \frac{M_n}{n!}R^n < \infty$, then f(x) is the taylor series.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function (i.e. assume that the n-th derivative $f^{(n)}$ is defined on all of $\mathbb{R}$). Let $R$ denote the radius of convergence of the Taylor ...
4
votes
1answer
317 views

Taylor series for cosine around $\pi/3$

I need the Taylor-Series for $ f(x) = \cos(x) $ in $ a = \pi/3$: \begin{align*} f(x) &= \cos(x - \pi/3 + \pi/3) \\ &= \cos \left( x - \frac{\pi}{3}\right) \cos\left(\frac{\pi}{3}\right) - ...
2
votes
0answers
65 views

Taylor series $\frac{\sin x}{x}$ convergence

I needed the Taylor series for $f(x) = \frac{\sin x}{x}$ in $a = 0$. I started with $ f(x) = \frac{1}{x} \cdot \sin(x) $, used the existing $sin$ Taylor series and multiplied by $\frac{1}{x}$: $$ ...
0
votes
1answer
41 views

Convergence of an analytic function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a smooth function. Let $R$ be the radius of convergence of the Taylor series centered at $a.$ For each $n \in \mathbb{N},$ let $M_n= \sup\{f^{n}(t) : t \in ...
0
votes
0answers
38 views

Taylor polynomial of $\sin(x)$

It is asked to construct the Taylor Polynomial $p_n$ (polynomial of order n) of the function $\sin (x)$, defined in $(-1,1)$ around 0. Also, I need to decide if $p_n$ uniformly converges to sine in ...
0
votes
2answers
48 views

Estimate Interval of Validity of $1-\frac{x^2}{2}$ for $\cos(x)$

I have been struggling with the following problem and was wondering if anyone could provide some insight or suggestions: Use $1-\frac{x^2}{2}$ as an approximation to $\cos(x)$, with an error not ...
3
votes
2answers
357 views

Sine taylor series

I'm pretty convinced that the Taylor Series (or better: Maclaurin Series): $$\sin{x} = x -\frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$ Is exactly equal the sine function at $x=0$ I'm also pretty sure ...
0
votes
1answer
79 views

exp(x) for imaginary numbers

Well, I know how to get the $e^x$ function polynomial expansion, but how do I know that this is also valid for imaginary numbers, like $i\pi$? I know that the ...
2
votes
1answer
101 views

Evaluate $I=\int_{\gamma (0,1)}\frac{zdz}{(z^2-4z+1)^2}$ using Taylor's Theorem

I've shown that $f(z)=\frac{z}{(z^2-4z+1)^2}$ is holomorphic apart from at points $\alpha=2-\sqrt3$ and $\beta=2+\sqrt3$ and that the talyor coefficient of $g(z)=\frac{z}{(z-\beta)^2}$ centred at ...
2
votes
1answer
284 views

Help find the MacLaurin series for $\frac{1}{e^x+1}$

What is the MacLaurin series up to $x^4$ for $\frac{1}{e^x+1}$? My Attempt: $$\begin{align} \frac{1}{e^x+1} &=(1+e^x)^{-1} \\ &\approx 1 -e^x+(e^x)^2-(e^x)^3+(e^x)^4 \\ \end{align} $$ Since ...
1
vote
0answers
39 views

Taylor series convergence

$$f(z)=\int^z_0 \frac{\zeta-\sin(\zeta)}{\zeta^2+4} \, d\zeta$$ I am supposed to find the convergence radius of its Taylor series at point $a=2$. I can find the radius in simple cases by finding ...
0
votes
0answers
82 views

Order of accuracy of the following approximation?

This is a computational analysis class, the answer shouldn't be complicated. Given the ODE: $$\begin{cases}y'=f(t,y)\\y(t^0)=y^0\end{cases}$$ What would be the order of accuracy (using Taylor ...
0
votes
1answer
215 views

What is the difference between Taylor series and Laurent series?

Can someone intuitively describe what is the difference between Taylor series and Laurent series? Also, what is the most general formula for both?
2
votes
1answer
53 views

Finding Taylor approximation for $x^4e^{-x^3}$

I'm trying to find Taylor approximation for the function: $$x^4e^{-x^3}$$ I started taking the first, second, third, etc. derivatives but the expression for it seems to explode with terms. I was just ...
2
votes
1answer
425 views

Why is$ (1+\frac{1}{n})^n=e$ when n goes to infinity? [duplicate]

Why is $\lim\limits_{n\to\infty}(1+\frac1n)^n=e$? I think it involves $\sum\limits_{n=0}^\infty\frac1{k!}=e$ but not sure how to get from one to the other.
1
vote
1answer
334 views

Numerical analysis Taylor's method question: Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$.

Let $f(x)=\tan^{-1}(x)$ Let $P_n(x)$ be the $n$th Taylor polynomial for $f(x)$ about $x_0=0$ Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$. Is ...
0
votes
2answers
169 views

Taylor expansion with random variables $\frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}$

In Einsteins theory of relativity the kinetic energy of an object is given by the following formula $$E_k = \frac{mc^2}{\sqrt{1-\frac{v}{c}^2}} - mc^2$$ where m is mass of the object at rest v iss ...
0
votes
2answers
51 views

Geometric Series into Maclaurin Series

Expand 1/(1+x) into Maclaurin Series I found f(0)=1, f'(0)=1, f''(0)=2!, f'''(0)=3! and so on Therefore f^(k)(0)=k! so would the series centered at 0 be equal to x^k ? Just want to check to see if I ...
1
vote
2answers
248 views

Short way? Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$

Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$ Does there exist any short way? I have to calculate all partial dervatives. Is it?
1
vote
1answer
283 views

A problem related to mean value theorem and taylor's formula

I guess I need to use Taylor's formula and the mean value theorem. I have no idea except for them. Note: honestly, this is not homework. I am studying by myself. Suppose that ...
1
vote
0answers
123 views

About Hessian of distance function

I'm studying the comparison Hessian theorem and I not understand the following: Let $(M, \langle\ ,\ \rangle)$ be a complete Riemannian manifold. Given $o\in M$, define $r=dist(o, \cdot)$. Then, for ...
1
vote
1answer
43 views

Rearranging power series expansion to get parameter on denominator

How can we rearrange $$T=\dfrac{k V+g}{gk}\bigg(kT-\dfrac{1}{2}k^{2}T^{2}+\dfrac{1}{6}k^{3}T^{3}\bigg),$$ to get $$T=\dfrac{2V/g}{1+k V/g}+\dfrac{1}{3}k T^{2}$$ ?
2
votes
0answers
164 views

Proving Lagrange's Remainder of the Taylor Series

My text, as many others, asserts that the proof of Lagrange's remainder is similar to that of the Mean-Value Theorem. To prove the Mean-Vale Theorem, suppose that f is differentiable over $(a, b)$ ...
1
vote
2answers
39 views

How to evaluate binomial coefficients when $k=0$ and $1\geq|n|\geq0$

So I am doing some taylor series which boil down to the geometric series, for which I then need to evaluate various binomial coefficients. I have always used $n\choose k$ $=\frac{n!}{(n-k)!k!}$ to do ...