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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117 views

Taylor Series Substitution $e^{x^2-1}$

If I'm using substitution to find a Taylor series about $x=1$ for $e^{x^2-1}$, but I'm given the Maclaurin series for $e^x$, how come the fact that the Taylor series is about $x=1$ doesn't matter when ...
2
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3answers
291 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.
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0answers
37 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 ...
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1answer
43 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 ...
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2answers
32 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 ...
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1answer
113 views

Proving Remainder of Taylor Series of 1/(1-x) approaches 0

It is well known that the Taylor (Maclaurin) series of $f(x) = \frac{1}{1-x}$ is $\sum_{n=0}^\infty x^n$ on $(-1,1)$. I am having difficulty proving the equality of these two. The error term is ...
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1answer
31 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 ...
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2answers
60 views

Using Taylor's Theorem to show $|x-tan(x)|\leq 1/300$ for $0\leq x \leq 1/10$

Using Taylor's Theorem deduce that for $0\leq x \leq 1/10$ $|x-tan(x)|\leq 1/300$ So my attempt; to get the taylors theorem about $x_0=0$ $f(x)=x-tan(x)$ $f'(x)=1-sec^{2}(x)$ ...
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2answers
87 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 ...
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1answer
68 views

Is there closed form for $(1-p)(1-p^2)(1-p^3)…$ or its Taylor expansion?

I was considering the following problem: Somebody uses a backup for something (e.g. backups a file) and the backup is equally reliable as original storage. The storage is not perfectly reliable and ...
3
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1answer
168 views

TaylorSeries of complete elliptic integral of the first kind

I want compute $K(k)$ as a Taylor Series; $k\in\mathbb{R}$ and $\vert k \vert < 1$. Can someone help me? $$ K(k):= \int^{\frac{\pi}{2}}_0 \dfrac{dt}{\sqrt{1-k^2 sin^2t}} $$ Results so far: $$ ...
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0answers
21 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 $x$ ...
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3answers
88 views

How to find the full Taylor expansion of the following:

I need to find the full Taylor expansion of $$f(x)=\frac{1+x}{1-2x-x^2}$$ Any help would be appreciated. I'd prefer hints/advice before a full answer is given. I have tried to do partial ...
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1answer
47 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 $a = 10$
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2answers
62 views

Can a function be approximated by finite number of Taylor expansion terms outside of disk of convergence?

Suppose we have a finite number of terms for Taylor expansion of a conditionally convergent function. For example, $f=\frac1{1-x}$ with expansion $f=\sum_{n=0}^\infty x^n$. This expansion diverges ...
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1answer
46 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 ...
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1answer
71 views

Find the Taylor polynomial of degree 4 for cos(x), for x near 0

I am self studying calculus and I need help solving a Taylor Series problem. 1a) Find the Taylor polynomial of degree 4 for cos(x), for x near 0: I think the answer would be: ...
2
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1answer
43 views

A question on Taylor expansion/approximation

Suppose we are given a continuos function $f(x)$ where $x \in [0,2]$, and the function $f(x)$ is $n$-th-order differentiable, for $n \in \mathbb{N}$ and $n>2$. Besides, we know that these ...
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1answer
288 views

Problem with Taylor (asymptotic) expansion of hyperbolic functions at infinity

(Note: I chose a general title, because I believe this discussion will be applicable to all other hyperbolic functions having an asymptote at infinity, but I will specifically be focusing on ...
3
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1answer
198 views

Approximating $\sqrt{101}$ using Taylor series methods

I'm trying to approximate $\sqrt{101}$ using the Taylor series for the function $f(x)=\sqrt{x}$ centered at the point $x=100$. I need to obtain an approximation that is within $0.01$ of the correct ...
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0answers
48 views

Proof of lagrange inversion of taylor series

is there a proof for the lagrange inversion of taylor series? The formula is given in http://en.wikipedia.org/wiki/Lagrange_inversion_theorem#Theorem_statement The proof cannot be found in the ...
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3answers
60 views

Having trouble calculating approximations using Taylor polynomials

I have a problem to approximate $\sqrt{1.06}$ using a third degree Taylor polynomial. The way I learned was to pick a center that we would know the answer to that is close to the value we're trying ...
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1answer
44 views

Taylor expansion of ln(1+x)

Find the Taylor expansion of $\ln(1+x)$ around $x=0$. I calculated: $f'(0)=1, f''(0)=-1, f'''(0)=1$, etc. $$T(3)=f(0)+f'(0)(x-0)+f''(0)(x-0)^2+f'''(0)(x-0)^3=0 + 1x-\frac{1}{2}x^2+\frac{1}{6}x^3$$ ...
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2answers
61 views

Finding the Maclaurin Series for $\sqrt{1+x^2}$

I can't find the Maclaurin series for $\sqrt{1+x^2}$. Every time it try to find it I get the Maclaurin series for $\sqrt {1+x}$. Can someone explain it to me? Thanks!
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1answer
15 views

Taylor Polynomial Variable Question

When you have a polynomial that you set your function equal to in the taylor polynomial (centered around $x = a$) $$function = c_0+c_1(x-a)+c_2 (x-a)^2+...$$ why is your variable $(x-a)$. Oddly ...
3
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2answers
232 views

Taylor Series of $\tan x$

I found a nice general formula for the Taylor series of $\tan x$: $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$ where $B_n$ are the Bernoulli ...
0
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1answer
20 views

Give the Maclaurin series for $f(x)=(3+e^{-x})^2$ and find values of $x$ for which this series converges.

Given is: $f(x)=(3+e^{-x})^2$ so I write: $$e^x=\sum_{n=0}^\infty \frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$$ $$e^{-x}x=\sum_{n=0}^\infty ...
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2answers
74 views

Maclaurin series: $\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^7}{7!}+\frac{x^8}{8!}+\frac{x^{11}}{11!}+\frac{x^{12}}{12!}+…$

The Taylor series of a real or complex-valued function ƒ(x) that is infinitely differentiable at a real or complex number a is the power series $$f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} ...
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2answers
67 views

Proof of $(1-e^{ix})^{-1}$

In G.H. Hardy's book 'Divergent Series' there is a claim that $(1-e^{ix})^{-1} = \frac {1}{2} + \frac {1}{2} i \cot (\frac {1} {2} x) $ I, for the life of me, can't get past showing that ...
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1answer
47 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.) $f(z) = {\frac{z-1}{3-z}}$ ...
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1answer
111 views

Complex Taylor and Laurent expansions

Let $f(z):=\dfrac{1}{2-z-z^2}, z\in\mathbb{C}\setminus\left\{ {1, -2}\right\}$. i) Express $f$ in the form $\dfrac{A}{1-z}+\dfrac{B}{2+z}$. [Answer to this is $\dfrac{1/3}{1-z}+\dfrac{1/3}{2+z}$]. ...
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2answers
66 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
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1answer
89 views

Taylor expansion of a not easily differentiable function

Context: I'm trying to find the period of a simple pendulum. As is well known, if the initial angle is small the period is approximately constant. I'm trying to do a second order expansion. I have ...
3
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2answers
92 views

Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
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3answers
269 views

Take 2: When/Why are these equal?

This didn't go right the first time, so I'm going to drastically rephrase the query. As per this previous question, I am wondering if the two series ...
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0answers
57 views

The Flat Function

I have to write an essay on the flat function $$\text{flat}(x) = \begin{cases} e^{-\frac{1}{x^2}} & \text{for } x \ne 0 \\ 0 & \text{for } x = 0 \end{cases}$$ and I want to prove ...
2
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2answers
35 views

Basic doubt on Taylor's polynomial

I have a doubt about a general situation in where I am asked to calculate $f(x)$ with a certain precision. How can I compute the number of terms of the Taylor polynomial needed for that? For example ...
3
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2answers
221 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, ...
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1answer
980 views

Remainder term of Lagrange Interpolation Polynomial

Suppose $x_0,x_1,\ldots,x_n$ are $n+1$ distinct numbers in the interval $[a,b]$ and $f\in C^{n+1}[a,b]$. Then for each $x$ in $[a,b]$, there is a number $\xi$ in $(a,b)$ such that $$f(x) = P(x) + ...
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0answers
121 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
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1answer
30 views

Compostion of functions in taylor series

I was attempting to help answer a question, but I am curious about the following. Suppose we have an analytic function $f(z)\equiv\sum\limits_{k=0}^{\infty} a_k(z-z_0)^k $ on some suitable disk of ...
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1answer
97 views

Finding the Taylor Series of this function

I am trying to find a series expansion of the following function: $$\left(\frac{\log x}{x}\right)^n$$ I need hints or methods for going about doing this. Is it even possible? I am on to something ...
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0answers
35 views

How to prove a parallel $(u=u_0) $it self curvature?

My name is Gita, and I had aproblem with my math. and need help, I know that a parallel $C$ in a surface of revolution in $M$ be a geodesic if and only if $f'(u_0)=0$. and $C$ is non arc lenght ...
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1answer
42 views

differential equation as taylor series

Consider the equation $\frac{\mathrm{d} x(t)}{\mathrm{d}t} = g(x(t))$ , with $x(0) = x_0$, where g is function that admits derivatives of all orders.If the solution of the equation can be written as a ...
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0answers
13 views

Range of convergence for Taylor's series gf given that of g and of f

Are the following 2 points correct? Let $D_f$ denote the maximal domain for which the Taylor's series of $f$ converges. 1) If $D_g = \mathbb{R}$, then $f$ converges $\implies gf$ converges. 2) On ...
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2answers
64 views

Maclaurin Series for a natural logarithm

Can anyone please help me with this question? Find the Maclaurin series and the interval of convergence for $f(x) = \ln(1-7x^9)$ I thought the answer was $$\sum_{n=1}^{\infty} (-1)^n ...
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2answers
630 views

Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

To everyone: don't bother writing up another answer, i'm giving this bounty Antonio's answer. It just doesn't let me yet (24 hours delay). If you map the nth roots of unity $z$ with the function ...
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1answer
57 views

Taylor's Theorem and inequalities on some interval of the domain?

From the following form of Taylor's Theorem and assuming that $|f(x)|\le 1$ and $|f''(x)|\le 1$ hold on $[0,2]$, $$f(a+h) = f(a) + hf'(a) + (1/2)h^2f''(a+θh),$$ some application of Taylor's Theorem ...
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1answer
61 views

Range of convergence for Taylor's series (about 0) for e^(sin x)

Is there anything wrong with my method below? Also, is there an easier method? For $sin\,x = \sum^{\infty}_{k=0}\frac{(-1)^kx^{1+2k}}{(1+2k)!}$, $L_1 = \lim_{k\rightarrow\infty} \left| ...
3
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0answers
79 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...