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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2answers
80 views

Verify f'(x) = e^x

The following is a proof I wrote to prove that given $f(x)=e^x$, $f'(x)=e^x$. For this proof we must use the Taylor Series for $e^x$, $\sum\limits_{n=0}^{\infty}\dfrac{x^n}{n!}$. Since the derivative ...
2
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2answers
41 views

Almost Taylor's Theorem Proof through Integration by Parts

I ALMOST derived Taylor's theorem, which here is $f(x)=\sum_{n=0}^\infty\frac{(x-a)^nf^{(n)}(a)}{n!}$, where $a$ is some arbitrary constant. My attempt: $$f(x)+C=\int f'(x)dx$$ $$\int ...
1
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0answers
227 views

3Dimensional runge kutta and Euler method ( help to verify the idea and proposition)

Previously,i been discussing this idea with a tutor for sometime. However it turn out that the proof is not comprehensible.But now the derivation for the accuracy up to $O(h^2)$ can be understood. ...
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0answers
16 views

Is the following correct way of manipulating taylors series?

For $\sum^{\infty}_{n=1}\frac{(-1)^{n}\pi^{2n}}{4^n(2n+1)!}$. Let $x=\frac{\pi}{2}$, the series becomes ...
2
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2answers
38 views

How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$?

How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$? It should be associated with the geometric series. Setting $t=x-3,\ x=t+3$, then I don't know how to continue, could someone clarify the ...
1
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1answer
34 views

How to compute $\sin{(\pi x)}$ about $\frac12$ in taylor series?

The correct answer is supposed to be $\sum\frac{(-1)^n}{(2n)!}\pi^{2n}(x-\frac12)^n$ which I don't understand. Since the function is about $x=\frac12$, so $(x-\frac12)^n$ is good. But ...
4
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7answers
549 views

How do you create an alternating series with the sign being the same twice in a row?

I am working on a Taylor series question and I have created a series which alternates however, it does so in doubles. in other words it follows the following pattern: $x$, $x$, $-x$, $-x$, $x$, ...
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1answer
85 views

Are there only a few 'universally convergent' Taylor Series?

The Taylor series for $\sin(x)$, centered at any point, converges for all $x$. The Taylor series for $e^{x}$ and $\cos(x)$ do as well. Thus, taking an algebraic function of these (without division) ...
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3answers
40 views

How to compute $\lim\limits_{x\to 0}\frac{e^{x^2}-\cos{(2x)}-3x^2}{x^2\sin{x^2}}$ limits by using taylor series?

How to compute $\lim\limits_{x\to 0}\frac{e^{x^2}-\cos{(2x)}-3x^2}{x^2\sin{x^2}}$ limits by using taylor series? I think that we need to take every familiar taylor series (i.e. $e^x,\sin{x}$) and ...
2
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1answer
25 views

How to compute the following series using taylor expansion manipulation?

How to compute $\sum^{\infty}_{n=0} \frac{x^n}{(n+2)n!}$ and $\sum^{\infty}_{n=0}(-1)^n \frac{(n+1)x^{2n+1}}{(2n+1)!}$ using taylor expansion manipulation? $1.\sum^{\infty}_{n=0} ...
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2answers
88 views

Characterizations of $e^x$

I've been thinking about the following problem for a while: (AFAIK) the 'exponential function', $e^x$ can be characterized as the unique solution to the following differential equation with initial ...
2
votes
1answer
18 views

How to compute taylor series for $f(x)=\frac{1}{1-x}$ about $a=3$ and $f(x)=\sin{x}$ about $a=\frac{\pi}{4}$?

How to compute taylor series for $f(x)=\frac{1}{1-x}$ about $a=3$ and $f(x)=\sin{x}$ about $a=\frac{\pi}{4}$ ? For the first one, using substitution, let $t=x-3$, then $x=t+3$. Then ...
0
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0answers
15 views

Maclaurin polynomial expansion of $y$ about 1?

Consider the differential equation $\frac{dy}{dx}=2x+\frac{y}{x}$, where $\frac{dy}{dx}=1$ when $x=1$. Find the first three non-zero terms in the Maclaurin polynomial expansion for $y$ about ...
1
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2answers
37 views

Uncertain how the following step was accomplished.

I'm working through a book example that aims to find the first two nonzero terms of the Laurent expansion of $f(z)=\tan(z)$, about $z=\frac{\pi}{2}$. The substitution $z=\frac{\pi}{2}+u$ is made ...
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3answers
48 views

Use of Taylor series expansion to find second derivative for sixth order method

Use Taylor's expansion to derive sixth order method (i.e $\mathcal{O}(h^6)$) for approximating the second derivative ($f '' (x_0)$ ) for given sufficiently smooth function $f(x)$. I have this things ...
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1answer
26 views

Taylor Polynomial Approximations

I am asked to find a Taylor Polynomial approximation accurate to within $10^{-3}$ for the following function $$f(x)=\frac{1}{x+1}, x \in [-\frac{1}{2},\frac{1}{2}]$$ We know the Taylor expansion for ...
0
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2answers
27 views

When does Taylor series for g agree with g

For $g(x)=e^{-1/x^2}$ for x not equal to 0 and $g(0)=0$. How to show that the Taylor series for g about 0 agrees with g only at $x=0$? I know that the maclaurin series for g(x) is ...
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2answers
34 views

Check: Radius of Convergence of the Sum of these Complex Taylor Series

I just found the following Taylor series expansions around $z=0$ for the following functions: $\displaystyle \frac{1}{z^{2}-5z+6} = \frac{1}{(z-2)(z-3)} = \frac{-1}{(z-2)} + \frac{1}{(z-3)} = ...
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5answers
109 views

Maclaurin Expansion for $e^{e^{z}}$ at $z=0$

I need to find terms up to degree $5$ of $e^{e^{z}}$ at $z=0$. I tried letting $\omega = e^{z} \approx 1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots$, and then substituting these first few terms ...
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0answers
42 views

Taylor series for $\arctan x$

We use $\frac{1}{1+x^2} = \sum_{n=0}^\infty (-1)^nx^{2n}$, where $|x|<1$ and integration yields $\arctan x = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{2n+1}$. And by the ratio test this series ...
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1answer
36 views

Find Maclaurin series for integral of $e^{z^2}$

I need to find a Taylor Series expansion of $\displaystyle \int_{0}^{z}e^{\zeta^{2}}d\zeta$ around $z=0$, which shouldn't be hard enough. Except that I can only integrate term-by-term if the Taylor ...
5
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2answers
80 views

Examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$

How to examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$ for $a, b, c> 0$ using Taylor's theorem?
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3answers
36 views

Prove that the $c_{n}$ in $\frac{1}{1-z-z^{2}}=\displaystyle\sum_{n=0}^{\infty}c_{n}z^{z}$ satisfy a Fibonacci-like recurrence relation

I need to prove that the coefficients $c_{n}$ of the expansion $\frac{1}{1-z-z^{2}}\sum_{n=0}^{\infty}c_{n}z^{n}$ satisfy the recurrence relation $c_{0}=c_{1}=1$, $c_{n}=c_{n-1}+c_{n-2}$, by ...
0
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2answers
26 views

Taylor series for $\frac{1}{az+b}$ centered at $z=0$ by substitution

I need to find the Taylor series centered at $z=0$ (i.e., the Maclaurin series) for $\displaystyle \frac{1}{az+b}$, where $a,b \in \mathbb{C}$ and $b \neq 0$. I thought it would be good to start out ...
0
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1answer
42 views

Finding $f^{(12)}(0)$ with $f(x)=\log(e^{x^4}-2x^8)$

Here's how I proceeded: We have $f(x)=x^4+\log\left(1-2x^8e^{-x^4}\right),$ hence for all $x$ such that $-1\le2x^8e^{-x^4}<1$ the following holds: \begin{align} ...
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2answers
54 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
0
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1answer
31 views

Taylor expansion of $\sin(x-y)$

A question asks me to find the partial derivatives of $f: \mathbb{R}^2 \to \mathbb{R}$ with $f(x, y) = \sin(x-y)$ then asks me to give the taylor expansion of $f(\pi/2+h, k)$ in powers of $h$ and $k$ ...
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4answers
69 views

What is the general term for $e^x/(1-x)$

What id the taylor series expansion for $\frac{e^x}{1-x}$? I know that the series expansion for $e^x$ is the sum of $\frac{x^n}{n!}$ from $0$ to $infty$. But how can I account for the $1- x$ in the ...
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0answers
9 views

Expand or approximate entropy of a two-term Gaussian Mixture

Is it possible to create some expansion to approximate this $h(a)$ for $a>0$ near $a\rightarrow0$? $$N(x,v)\equiv\frac{1}{\sqrt{2\pi v}}e^{-\frac{x^{2}}{2v}}$$ $$ ...
0
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1answer
27 views

How do I work out the validity for a Maclaurin (power) series?

I cannot find the answer to this anywhere so I have decided to make a question. Given a Maclaurin series for a function, how can I quickly work out what the validity is for it? For example, ...
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2answers
71 views

Why Taylor Series or any other approximation method give us approximation of function? Why not give exact equivalent of function?

Lately I am started studying approximation of functions by polynomials and the need for approximation of functions? But what I failed to understand and books did not explain me is that why finding ...
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2answers
56 views

Maclaurin series of $x^3/(e^x-1)$

how would i taylor expand $f(x)=\frac{x^3}{e^x-1}$ around $x=0$? I was thinking of writing $\frac{x^3}{e^x-1}\approx\frac{x^3}{1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\dots}$ $~~~~~~~~= ...
12
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1answer
286 views

Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for ...
4
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6answers
3k views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
1
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2answers
50 views

Why does the taylor series of $\frac {1}{\ln x}$ have a non-infinite radius of convergence?

Shouldn't the taylor series of a function be equal to that function for any input value? Why does this not work for the taylor series of $\frac {1}{\ln x}$ when $|x| \gt 1$? Edit: I do mean the ...
1
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3answers
34 views

Taylor's series and ln

Can someone explain to me how to find the $\lim \limits_{x \to 3} \frac{\ln|4-x|}{x-3}$ using taylor's series. Can someone explain the proof of $\ln|4-x|$ to power series please
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1answer
24 views

Radius of Convergence of Taylor series without finding the series

How do you find the radius of convergence of a Taylor series for a function centered at point $z_0$ without actually finding the Taylor series? I know that we can use comparison test, ratio test or ...
2
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1answer
24 views

about truncated taylor expansions

I have a question about expanding $e^u$into a truncated taylor series where $u$ is itself a truncated Taylor series (in my example $u$ is expansion of $-\frac{\log(1+t)}{t}$, up to term $O(t^3)$), it ...
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1answer
26 views

Expanding $1/z$ about $z=-1$ using Taylor series vs Power Series

I need to expand $1/z$ about $z_0=-1$. I decided to do it using both methods, which don't agree. Using Taylor: Finding coefficients: $$f^{(n)}(z)=(-1)^n n!/z^{n+1} \Rightarrow f^{(n)}(-1)=-n!$$ ...
3
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2answers
35 views

Exponential Taylor series with $k$ step

It is well-known that $$\sum_{n=0}^\infty \frac{x^n}{n!} = e ^x$$ or $$\sum_{n=0}^\infty \frac{x^{2n}}{(2n)!} = \cosh x $$ My question is what we know about the sum for arbitrary $k \in \mathbb{N}$: ...
1
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1answer
39 views

Complex Taylor Series by substitution

I need to find the first few terms or so of the Taylor series centered at $z_0 = 0$ regarding these functions: a) $e^{z\sin z}$ b)$(1+z)^z = e^{z \ln (1+z)}$ c)$\cos (1 + z^3) $ d) $e^{e^z}$ ...
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0answers
15 views

Value of Taylor Approximation

I'm studying Trust Region Methods and there's one simple thing I'm finding it hard to wrap my head around. The premise of the method is to approximate $f(x)$ as a quadratic model: ...
2
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1answer
34 views

How do I expand this function around zero?

The function is $$ \sqrt{\frac{\sin(x)}{x}} $$ I need to expand it to the order $x^2$ around $0$. The solution is supposed to be: $$ 1-\frac{x^2}{12}+\mathcal{O}(x^4) $$ How do I proceed?
2
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3answers
42 views

$n$-th Term for Maclaurin Series

On a Calculus BC test I had this morning, I had to find the first five terms and the $n$-th term of the following function: $$ f(x) = x \cos(3x)$$ According to my instructor, I could've manipulated ...
24
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2answers
1k views

How local is the information of a derivative?

I have read it a thousand times: "you only need local information to compute derivatives." To be more precise: when you take a derivative, in say point $a$, what you are essentially doing is taking a ...
4
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2answers
203 views

Applying Taylor theorem on a linear map

I found the following in a stack of practice problems but had trouble dealing with it: Consider a linear map $A:C^\infty(\mathbb{R}^n)\rightarrow \mathbb{R}$ such that: If $f\in ...
2
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1answer
27 views

Taylor vs Laurent series - cosines and sines

In general, why do we say that the Taylor series of sines and cosines are also Laurent series despite of the power of $z$?
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0answers
31 views

Estimate sqrt(1.1) using Lagrangian formula and Taylor polynomials with error within 1/10^6

So I set f(x)=sqrt(1+x) and then went on to estimate the error for x=0.1 according to the Lagrangian formula will be f(n+1)(ξ)*0.1^(n+1)/(n+1!). I know 0<ξ<0.1 but I still cannot think of how ...
0
votes
1answer
28 views

Taylor expansion, problematic integrand

Consider $$f(z) = \int_0^z \frac{1-\cos\sqrt{t}}{t}\mbox{d}t $$ Find its Taylor series at $a=0$. I was thinking about looking at the integrand, from which we would have: $$\frac{1-\cos\sqrt{t}}{t} = ...
0
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0answers
43 views

Taylorseries of $\cos(x)e^x$

Lets consider $f:\mathbb R\rightarrow \mathbb R, f(x)=\cos(x)e^x$. I want to calculate the taylor-series around $x_0=0$ and I want to check if the taylor-series is equal to $f(x)$. The first ...