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.

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4
votes
2answers
59 views

Taylor expansion for Si(x)?

I want to find out what the Taylor expansion of $$F(x) = \int_0^x \frac{\sin(t)}{t} dt .$$ Am I wrong in saying that by the fundamental theorem of calculus, $F'(x) = sin(t)/t$? Should I continue ...
7
votes
2answers
121 views

Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$

Prove the following inequality: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ This should be solved using Taylor series. I tried expanding the left to the 5th degree and the right site to ...
1
vote
2answers
60 views

Maclaurin series for $f(x)=\frac{1}{1+x+x^2} $

What is the Maclaurin expansion of $f(x)=\dfrac{1}{1+x+x^2} $? Thank you! Edit: By multiplying both terms with $ (1-x) $ I got to $\dfrac{1}{1-x^3}-\dfrac{x}{1-x^3}$. Is it correct to transform ...
2
votes
3answers
93 views

$\lim_{x\to\infty} \frac{5\cdot5^x+3^x-4^x}{5^x +2^x+27\cdot9^x}$

How can I solve this limit. (Here $x$ belongs to natural numbers $\Bbb{N}$.) $$ \lim_{x\to\infty} \dfrac{5\cdot5^x+3^x-4^x}{5^x +2^x+27\cdot9^x}$$ My try: I tried using L'Hospital, expansions of ...
1
vote
3answers
73 views

Taylor expansion - what order would be preferred?

Let say you want to calculate the following limit: $$\mathop {\lim }\limits_{x \to 0} \left( {\frac{1}{{1 - \cos x}}\ln \left( {\frac{{\sin x}}{x}} \right)} \right)$$ Obviously, Taylor Expansion ...
1
vote
1answer
368 views

What's wrong in my Taylor series implementation in MATLAB?

I'm trying to code Taylor summation for a function in Matlab, I actually evaluate McLaurin making $x_0=0$, named a in this code after this notation: This is the code I've tried out so far: ...
2
votes
5answers
91 views

What does $a$ mean in Taylor series formula?

I'm trying to code the Taylor summation in MATLAB, being Taylor's formula the following: I've also seen $a$ denoted as $x_0$ in distinct bibliography. Problem is that I'm not sure how should I ...
6
votes
4answers
1k views

Using Taylor's Theorem to show that $\ln(1 + x^2) \leq x^2$

Can we show that if $\operatorname{abs}(x) \lt 1$, then $$\ln(1+x^2) \leq x^2\;,$$ using Taylor's Theorem? I am thinking of expanding it about $x=0$ but I got something like $$f(x) = -x^2 + ...
3
votes
2answers
526 views

Express $\sin nx$ and $\cos nx$ in terms of $\sin x$ and $\cos x$ respectively

What are the expansions of $\sin nx $ and $\cos nx$ in terms of $\sin x$ and $\cos x$ respectively? (here $n \in \mathbb N$). Maybe this is solved problem or there is new technique to ...
0
votes
1answer
36 views

Tangent and Taylor polynomials

We know that this series $x+ \frac{x^3}{3}+\frac{2x^5}{15}+\ldots$ is convergent in $|x|\lt \pi/2$, furthermore it converges to $\tan(x)$. I would like to know if we restrict to finite terms of this ...
1
vote
1answer
44 views

Asymptotic Expansion in zero of $\frac{1}{\ln(1+x)}$

On wolfram the expansion is: $$\frac {1}{x} + \dfrac{1}{2} ...\,.$$ But I don't understand from where it outside comes the $\frac{1}{2}$ thanks
0
votes
0answers
169 views

taylor series of $\sin t/t$

I know that the taylor series expansion around zero of $\sin t/t$ is: $\sum_{k=0}^{\infty} (-1)^k \frac{(t^{2k})}{(2k+1)!}$ , I need to find its radius of convergence. I saw a few solutions that claim ...
0
votes
2answers
38 views

Maclaurin Expansion of $\frac{x}{\sqrt{4-2x}}$

Maclaurin Expansion of $\frac{x}{\sqrt{4-2x}}$ up to order 4. I really don't know how to do this, I can't find a helpful Maclaurin Series in my formula book to help me. I want to do $x(4-2x)^{-1/2}$ ...
1
vote
0answers
38 views

Landau identification

Calculate taylor series for $x\rightarrow +\infty$ at the higher order allowed by the approximation present in it. $$ \sqrt{x^6+x^5-2x^3+O\left(x^2\right)} $$ I made this: $$ ...
0
votes
2answers
35 views

Tayors series exansion of $(1 - x )^n$ where $0<x<1$ and $n \ge 0$

I want to find Taylor's series or Maclaurin's series expansion of the following. $$(1 - x)^n \ \text{ where }\ \ 0 < x < 1 \text{ and }\ n \ge 0$$ will it be same as that for $$(1 + x)^n ...
1
vote
1answer
30 views

Expanding One Function in Powers of Another

One sees here that it is possible to expand $f(x) = 2x^3 + 7x^2 + x - 6$ in powers of $x - 2$ by taylor expanding $f(x) = f(x - 2 + 2) = f(2 + h)$ about $2$, and this idea can be used in deriving the ...
12
votes
4answers
784 views

Taylor expansion of $(1+x)^α$ to binomial series – why does the remainder term converge?

For $α ∈ ℝ$ the function $g_α \colon B_1(0) → ℝ, x ↦ (1+x)^α$ is $C^∞$ and $g_α^{(n)}(x) = n! \tbinom{α}{n}(1+x)^{α-n}$, where $\tbinom{α}{n} = \frac{α(α-1)\cdots(α-n+1)}{n!}$ is the generalized ...
-1
votes
1answer
49 views

Taylor Theorem conceptual question

Need to solve following-- Let $$ f(x)=\begin{cases} 0& -1\le x\le0\\ x^4& 0\lt x\le 1\end{cases} $$ IF$$ f(x)=\sum_0^n\frac{f^{(n)}(0)}{n!}(x)^n + \frac{f^{(n+1)}(c)}{n+1!}(x)^{(n+1)}$$ is ...
1
vote
1answer
53 views

Evaluate the series

Let $f:(-1,1)\to \mathbb{R}$ defined as $$f(x)=\frac{x^2}{1-\cos x}$$ for $x\neq 0$ and $f(0)=2$. If $f(x)=\sum_{n=0}^{\infty}a_nx^n$ is the Taylor expansion of $f$ for all $x\in(-1,1)$, then ...
2
votes
1answer
52 views

The 7-th derivative of $ x^3 \cdot\tan(2x) $ is this right

I have to find $y^{(7)}\left(0\right)$ of $y(x)=x^3\cdot\tan{(2x)}$ So my idea was to use Taylor expansion for $\tan(2x)$ to the $7$-th element and then multiply the hole thing by $x^3 $ and then ...
3
votes
2answers
148 views

The integral $\int\ln(x-\ln(x))~dx$

The integral $f(y)=\int_0^y\ln(x-\ln(x))~dx$ is on my mind. I'm not sure if this has a closed form? Maybe we need to use the lambert-W function to solve this one? If it cannot be done in closed ...
3
votes
2answers
71 views

Coincidence of $x-\frac{x^3}{6}$ and $\sin x$ in an interval

Plotting $f(x)=x-\frac{x^3}{6}$ and $g(x)=\sin x$, one can see that these two function are coincide in an interval $I\subset(-\frac{\pi}{2},\frac{\pi}{2})$. On the other hand, Taylor series for $\sin ...
3
votes
1answer
55 views

Taylor series- help! [closed]

$f(x)$ is twice differentiable function in $[0,1]$. we know that: $f(0)=0$, $f(1)=1$, $f'(0)=f'(1)=0$. show that there exists a point $c$ such that $\left|f''(c)\right|\ge 4$ Thanks in advance
0
votes
2answers
1k views

Taylor series for $\cot x$

Hi guys could you show me how to do the expansion of the Taylor series of $\cot x $ at the point $x=0$. My idea was to use $\dfrac{\cos x}{\sin x} $ and I want to expand it to the second term because ...
0
votes
2answers
45 views

how can I sove approximation evaluation of this integral?

$$\int_{-1}^{0}\sin(e^{x})\,dx $$ approximation of this formula up to difference(error) $1/5000$ Because of the error size $1/5000$ , I think it's solved by taylor expansion.
2
votes
1answer
109 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 ...
3
votes
1answer
481 views

Proof of the correctness of Taylor series

I am looking at the proof provided on the wiki page for taylor series http://en.wikipedia.org/wiki/Taylor%27s_theorem#Proof_for_Taylor.27s_theorem_in_one_real_variable One of the proof provided is ...
3
votes
1answer
64 views

Mistake in Taylor expansion?

Given: The first derivative of $\tan x$ is $1/\cos^2 x$ So the derivative of $\tan x$ when $x=0$ should be $1$. This derivative times $x$ should be a term in the Taylor expansion (the term then being ...
0
votes
3answers
395 views

solve the initial value problem ,by Taylor's method of order $N=3$

solve the initial value problem ,by Taylor's method of order $N=3$ $y'(t)=ty(t)+(1-t)e^t,0\le t\le 2,y(0)=1$ with an accuracy of $5 \times10^{-3}$ first we consider the taylor expansion of $e^x$ $ ...
3
votes
2answers
483 views

How to calculate Taylor expansion of $\cos(\sin x)$

I know that Taylor expansion of $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + O(x^6)$ and that of $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - O(x^7)$. But how do I calculte the Taylor Expansion ...
3
votes
1answer
79 views

Find all $\alpha \in \mathbb{R}$ such that $\int_0^\infty \sin(x^\alpha)dx$ converges.

Find all $\alpha \in \mathbb{R}$ such that $\int_0^\infty \sin(x^\alpha)dx$ converges. There is an answer here that differs from mine (they claim for $-\infty<\alpha<-2$ and ...
0
votes
1answer
46 views

Proving this Taylor-esque expansion for a $C^2$ function vanishing at 0 and 1

I am trying to prove the following (which I think is true!): if $f:[0,1]\rightarrow \mathbb{R}$ is twice continuously differentiable and $f(0)=0=f(1)$, then for every $x \in (0,1)$ there exists $\xi ...
1
vote
1answer
64 views

Taylor series expansion - application

I am working on the following: Let $f : \mathbb C \to \mathbb C$ be analytic. Suppose for all $z \in \mathbb C$ hold $f(2z) = 4f(z)$ and $f(1) = 1$. Then $f(z) = z^2$ for all $z \in \mathbb C$. I ...
2
votes
1answer
65 views

Laurent series for $\frac{z}{z+1}$ when $1<|z|<\infty$

Calculate the Laurent series for $\displaystyle\frac{z}{z+1}$ when $1<|z|<\infty$. There is really no singularity here, right? Can I just use a Taylor series, or what should I do?
1
vote
2answers
60 views

Does this limit imply that a function is “close” to Lambert W?

Suppose I am given the following limit involving function $f(n)\geq 0$: $$\lim_{n\rightarrow\infty}\log n-f(n)-\log f(n)=c$$ where $c$ is a constant. I am wondering if that implies that $f(n)$ is ...
1
vote
2answers
63 views

Estimate $\int_{-1}^{0}\sin(e^x)dx$ with error less than $\frac1{5000}$.

Let $f(x)=\sin (e^x)$ then the taylor polynomial of degree 2 at $x=0$ is $P_2(x)=\sin 1+(\cos1)x+\frac12(\cos1-\sin1)x^2$. I want to estimate $\int_{-1}^{0}\sin(e^x)dx$, using $P_2(x)$, with error ...
3
votes
1answer
189 views

Convergence radius of $\sqrt{\cos(z)}$

Compute the first 3 non zero terms of the Taylor expansion of $\sqrt{\cos(z)}$ at $z=0$ and determine its convergence radius, considering only the principal branch of the square root. I've computed ...
0
votes
2answers
81 views

Error estimation for $f(x)=\sin \sqrt{x}$

Let $f(x)=\sin \sqrt{x}$, then $f'(x)=\frac1{2\sqrt{x}}\cos \sqrt{x}$ and $f''(x)=-\frac1{4x\sqrt{x}}\cos \sqrt{x}-\frac1{4x}\sin \sqrt x$. Thus the Taylor polynomial of degree 2 at $x=\frac{\pi^2}9$ ...
2
votes
2answers
153 views

Product of two Taylor series

I have the following product of two Taylor series: $$f(x)g(x)=\frac{1}{z-1}\frac{1}{z-2}=\sum_{n=0}^{\infty} z^n \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} z^n$$ I wanted to know 2 things: 1st. How can ...
1
vote
2answers
36 views

Derivative of a Taylor series

I have a question about when we compute the derivative of a series. If the original series converges inside a region $R$, must its derivative also converge on the same region $R$?
1
vote
3answers
165 views

Taylor series for $e^z\sin(z)$

How can I write the Taylor series for $e^z\sin(z)$ at $z=0$ without making the procedure too complicated? Isn't there an easier way than to compute it's derivatives and find a pattern?
1
vote
0answers
113 views

The remainder of Taylor (Maclaurin) series of $\cos(x)$

Something is bothering me with the remainder of the Taylor (Maclaurin) series of $\cos(x)$. The formula of $a_n$ is $(-1)^k \frac{x^{2n}}{(2n)!}$. By Leibniz Theorem, $r_n<a_{n+1}$ which is, ...
2
votes
1answer
96 views

Taylor series in order to find the approximate antiderivative of a function

Somewhat inspired by this question about antiderivatives, I started to check whether or not that function had an elementary antiderivative. Then, after checking with Maxima, it struck me that, by ...
0
votes
3answers
57 views

I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator.

Let $f(x)=\ln x$. Then the Taylor polynomial of degree 2 at $x=e$ is $P(x)=1+\frac1e(x-e)-\frac1{2e^2}(x-e)^2$ I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator. ...
1
vote
3answers
77 views

General form for the series expansion of $e$

I've found a lot of series expansions of the Napier's constant. I was wondering if a general form for this could be devised. They all have a trend and similarities. I've been trying but I've been ...
2
votes
4answers
109 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 ...
2
votes
4answers
81 views

$\ln\left(1+x\right)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dots$ when $|x|<1$ or $x=1$.

$\ln\left(1+x\right)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dots$ when $|x|<1$ or $x=1$. Why is the restriction $|x|<1$ or $x=1$? I know from Wikipedia that it is because out of this restriction, the ...
3
votes
0answers
53 views

Characterization of functions with fractional expansion near zero

I would like to understand if it is possible to completely characterize real-valued functions with an expansion of this type: $f(x)=f'(0)\cdot x + o(x^{\alpha})\qquad \alpha \in (1,2)$ I am not ...
3
votes
1answer
807 views

Taylor series convergence for $e^{-1/x^2}$

Consider the Taylor series for $e^{-1/x^2}$ around $0$: $$e^{-1/x^2}=1-\dfrac{1}{x^2}+\dfrac{1}{2!x^4}-\dfrac{1}{3!x^6}+\ldots$$ For which $x$ does the series on the right converge to $e^{-1/x^2}$?