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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Plank's first law expansion

I'm getting a little stuck on this question. The question is: show that for $KT \gg h\omega$, the first law of Planck: $\displaystyle U =\frac{h\omega}{e^{(h\omega/KT)}-1} \approx KT - ...
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2answers
43 views

Taylor Polynomials — $\cos(x)$ [closed]

Show that $$\forall x : \cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots+\frac{(-1)^nx^{2n}}{(2n)!}$$ I know that this is true because it is one of the most common Taylor polynomials. ...
1
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2answers
43 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} - ...
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2answers
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Remainder form of Taylor polynomial at $x_0$: $ \frac{1}{n!}f^{(n)}(x_0 +\theta(x-x_0))(x-x_0)^n$ with $\theta \to \frac{1}{n+1}$ as $ x \to x_0$

If the function $f: \mathbb R \to \mathbb R$ is $n+1$ times differentiable at $x_0$ and $f^{(n+1)}(x_0) \neq 0$, then a form of the remainder in Taylor's Formula is supposedly $$r_n(x_0;x) = ...
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1answer
1k views

Prove Taylor expansion with mean value theorem

On http://vapour-trail.blogspot.com/2006/03/brief-explanation-of-taylor-series-via.html one can find an hint at how to derive Taylor expansions from the mean value theorem. The process goes as ...
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2answers
47 views

Find the Taylor series generated by f at x=a.

$f(x) = \frac 1 {9 - x}, a = 3$. The answer in the book is $$\sum_{n = 0}^{\infty} \frac{(x - 3)^n}{6^{n + 1}}$$but I'm not sure how to get the above.
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1answer
27 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
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1answer
30 views

How is the Harris Corner detector derived from a Taylor Expansion?

It looks more like an assumption about local linearity of the function when the shifts u and v are small. This description of ...
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3answers
47 views

Convergence of Taylor Series

My professor made this claim about Taylor Series convergence in my Complex Variables class and I am still not entirely convinced (he said it's explained in the textbook and textbook states, "we will ...
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0answers
24 views

Somehow “mirroring” the Taylor-expansion of some $g(x)$

In proceeding with an older discussion of a summation-procedure for divergent series using the matrix of Eulerian numbers I came to a rather general formulation but cannot/do-not-know-how-to make ...
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2answers
231 views

Derivation 9.97 in Jaynes' Probability Theory

In page 298 of Jaynes' Probability Theory: the Logic of Science, equation (9.97), Jaynes says: We expect that, if hypothesis $H$ is true, then $n_k$ will be close to $np_k$, in the sense that the ...
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1answer
65 views

Why does $\arctan(\frac{\tan \theta}{2}) \approx \frac{1}{2 - \theta} - \frac{1}{2 + \theta}$ for small $\theta$?

In answering this question, I needed to show that $\arctan \left( \frac{\tan \theta}{2} \right) \approx \frac{\theta}{2}$ when $\theta$ was small. So, naturally, I computed the first few terms of the ...
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1answer
23 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 ...
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0answers
28 views

Analytic Function Derived From Recursive Reverse Taylor Series?

Given the following recursive relation: $a_0 = 1,$ $a_n = a_{n-1}(p-2q)+2(-p)^n$ is there a simple function that has this as its Taylor series, i.e. $f(x) = \sum_{n=0}^{\infty} \frac{a_n}{n!}x^n$ ...
0
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1answer
32 views

Maclaurin series and taylor

Im trying to find the first four terms of Maclaurin series of $\space0.15t^2$ and evaluate $$\int_0^1 e^{-0.15}t^2 dt $$ Please this is a revision question. How do i go about it?
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2answers
66 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
77 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
32 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
41 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
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 ...
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1answer
36 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
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 ...
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0answers
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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 ...
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2answers
36 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
55 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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4answers
2k views

Simplest proof of Taylor's theorem

I have for some time been trawling through the Internet looking for an aesthetic proof of Taylor's theorem. By which I mean this: there are plenty of proofs that introduce some arbitrary construct: ...
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1answer
62 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 ...
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1answer
86 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
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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
82 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
43 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
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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
35 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
42 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: ...
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1answer
34 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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2answers
234 views

Euler's identity in matrix form

I assume everyone is familiar with the famous mathematical identity due to L. Euler: $$ e^{i \, \pi} + 1 = 0,$$ where $i^2 = -1$ and $e$ is the base of natural logarithms. I was wondering if this ...
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1answer
93 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 ...
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1answer
75 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
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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
37 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
32 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
39 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
13 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
86 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 ...
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1answer
18 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
68 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
66 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
24 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
75 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
53 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 ...