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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Computation of the remainder term on a Taylor expansion using contour integrals

I am not really used to the methods of complex analysis, I would like to know for basic monotonic functions like exp(x), log(x), sqrt(x), powers (x^n) and trigonometric functions defined on an real ...
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104 views

Taylor Polynomials, Why only Integer Powers?

So It seems that the definition of polynomial is that is is raised to an integer power, but why is this necessary? My question mainly arises from a proof of the solution to the Hydrogen atom in ...
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69 views

Radius of convergence for Taylor series?!

Given is: $f(x) = \frac{\sin x}{x} $ I need the Taylor series in $a = 0$, so: $$T(x,0) = \frac{1}{x} \sum_{n=0}^\infty ((-1)^n* \frac{x^{2n+1}}{(2n+1)!} ) = \sum_{n=0}^\infty (-1)^n * ...
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88 views

Lagrange Taylor remainder: can we choose $t^*$ continuously?

The Taylor theorem with Lagrange remainder tells us that for $f: \mathbb{R}^n \to \mathbb{R}$ twice differentiable (we can assume $C^2$ if we like), $$f(y) - f(x) = \left\langle \nabla f(x), y-x ...
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61 views

Weierstraß approximation on the real line

First of all: I am aware of the thread Weierstrass approximation does not hold on the entire Real Line. My question is just that if we have a function like $sin(x)$ that can be approximated by its ...
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147 views

An exception to Taylor Series

According to Taylor Series, $$f(x) = \sum_{n=0}^\infty \dfrac{f^{(n)}(a)}{n!}*(x-a)^n $$ However, $\dfrac{1}{x}$,$\dfrac{1}{x^2}$, etc. are not applicable. I tried to do the following: $$ ...
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455 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 ...
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532 views

Taylor's theorem for vector valued functions

I'm reading about linear and nonlinear programming and on one page I have the following statment (I have highlighted the areas where I have problems and drawn questions for them in the bottom of it): ...
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97 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 ...
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116 views

Taylor/Maclaurin Series Exam Question.

Show that if x is small compared with unity, then $$f(x)=\frac{(1-x)^\frac{-2}{3}+(1-4x)^\frac{-1}{3}}{(1-3x)^\frac{-1}{3}+(1-4x)^\frac{-1}{4}}=1-\frac{7x^2}{36}.$$ I've expanded all the brackets ...
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304 views

Taylor expansions at $x=\infty$

How do you expand, say, $\frac{1}{1+x}$ at $x=\infty$? (or for those nit-pickers, as $x\rightarrow\infty$. I know it doesn't strictly make sense to say "at infinity", but I think it is standard to say ...
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1answer
286 views

Using partial fraction for $\cot \pi z$ to compute infinite sum

I want to compute the values $\sum_{n=1}^\infty \dfrac{1}{n^2}$ and $\sum_{n=1}^\infty \dfrac{1}{n^4}$ and $\sum_{n=1}^\infty \dfrac{1}{n^6}$ by comparison to the partial fraction development of $\cot ...
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Practical Use of Series Expansion at $x=\infty$

Asking WolframAlpha on certain functions, it happens that you get a series expansion at $\infty$. Thinking of the expansion as an approximation of the function in the vincinity of a point $a$, like in ...
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183 views

Taylor series with Fibonacci coefficients

Let $\{a_n\}$ be the Fibonacci numbers given by $a_0=0,a_1=1,a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$. Prove that $f(z)=a_0+a_1z+a_2z^2+\ldots$ is a rational function, and determine which rational ...
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346 views

Maclaurin series for $e^x +2e^{-x}$

I'm currently stuck on the question regarding the Maclaurin series for $e^x +2e^{-x}$ I've found that the power series representation for it is $$\sum_{n=0}^\infty \dfrac{x^n + 2(-x)^n}{n!}$$ ...
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54 views

Taylor's formula in several

What i'm puzzling about is this: I use the following form of Taylor's formula with integral remainder term: For a smooth function $\varphi$ it holds that $$ \varphi(x+h) = \varphi(x) + \int_0^1 ...
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390 views

Adding two Power series or Maclaurin sums together and their radius of convergence

Say you have two power series. One of them has ROC of 2, and the other one has an ROC of 4. If you add the two series together is the ROC ALWAYS the lesser ROC? It seems to be a trend I've noticed, ...
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67 views

Taylor/Maclaurin Series

Show that if x is small compared with unity, then $$f(x)=\frac{(1-x)^\frac{-2}{3}+(1-4x)^\frac{-1}{3}}{(1-3x)^\frac{-1}{3}+(1-4x)^\frac{-1}{4}}=1-\frac{7x^2}{36}$$ In my first attempt I expanded all ...
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179 views

Fractional Derivative of a Taylor Series?

I have a function defined only by it's taylor series: $f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!}x^k$ Obviously, integer derivatives can be defined as $\frac{d^n}{dx^n} f(x) = \sum_{k=0}^\infty ...
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90 views

McLaurin series of complex function

I've got a function $g(z) = \frac {(1-z)(e^z + e^{-z})}{e^z - e^{-z}}$. I have to find coefficients $c_0, c_1, c_2, c_3$ of McLaurin's series of function $g$ (which is $\sum_{n=0}^{\infty} c_n z^n ...
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3k views

Multivariate Taylor Expansion

I am in confidence with Taylor expansion of function $f\colon R \to R$, but I when my professor started to use higher order derivatives and multivariate Taylor expansion of $f\colon R^n \to R$ and ...
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110 views

how to find the asymptotic expansion of the following sum:

I need to determine an asymptotic expansion when $q \rightarrow 1$ of the sum $$S(q)=\sum_{n=0}^{\infty} \frac{q^n}{ (q^n + 1)^2 }.$$ Numerical computations suggest that $S(q)\sim\frac{c}{|q-1|}$ ...
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201 views

How to know if a MacLaurin/Taylor Series expansion is good?

This question is motivated by this question. So, given $\frac{1}{e^x + 1}$, the 4th order MacLaurin series $1 -e^x+(e^x)^2-(e^x)^3+(e^x)^4$, although correct in terms of the algebra manipulations, is ...
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321 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 ...
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Whats wrong in following attempt to write Hermite polynomials in terms of hypergeometric function?

Let's have Hermite polynomials: $$ e^{2tx - t^{2}} = \sum_{n = 0}^{\infty}H_{n}(x)\frac{t^{n}}{n!}. \qquad (1) $$ I need to write it in terms of confluent hypergeometric Kummer function for index $n = ...
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40 views

Maclaurin series for $ f(x)=a^x$

My friend is having trouble with these two questions on his homework. I want to help him out but I am not 100% sure how to do these. I took Calculus 3 a while back so its all old memory to me! ...
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309 views

Taylor Series Approximation for degree k Taylor polynomial?

Let $T_k(x)$ be the degree $k$ Taylor polynomial of the function $f(x)=\sin(x)$ at $a=0$. Suppose you approximate $f(x)$ by $T_k(x)$. If $|x|\le 1$, how many terms are needed (that is, what is $k$) ...
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100 views

Differentiating second order term of Taylor polynomial (multivariable)

I am trying to derive Newton step in an iterative optimization. I know the step is: $$\Delta x=-H^{-1}g$$ where H is Hessian and $g$ is gradient of a vector function $f(x)$ at $x$. I also know the ...
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49 views

Version of Taylor: $F(x+h)-F(x) = \left \langle \int_0^1 \nabla F(x+th)dt, h \right \rangle.$

My teacher claimed without proof that Taylor's theorem with remainder implied that for a suitable function $F: \mathbb{R}^n \to \mathbb{R}$, $$F(x+h)-F(x) = \left \langle \int_0^1 \nabla F(x+th)dt, h ...
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114 views

Compute the 10th derivative

$f(x) = (\cos(5x^2) - 1 )/ x^2 $ at $x = 0$ We were given the hint to use the MacLaurin series for f(x). I get how to do it if it was just $\cos(5x^2)$ but what would I do with the other values in ...
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936 views

Find the Maclaurin series of the function f(x) = (7 x^2) sin (2 x)

Find the Maclaurin series of the function $f(x) = (7 x^2) sin (2 x)$ $(f(x) = \sum_{n=0}^{\infty} c_n x^n) $ That is what is given on the question, we have to fill in 5 blanks $c_3$ to $c_7$ The ...
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146 views

Gamma Type Integral

I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce: ...
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48 views

using Taylor's formula in a proof

Prove that $1+\frac{1}{n} < e$ for all $n$ in the natural numbers. How does this connect to Taylor's formula? I know that $e^x > 1+x$ for $x>0$, but then where does Taylor's formula come in ...
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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 ...
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118 views

What is the relationship between saying “a Taylor series converges for all $x$” and “a Taylor series converges to a function, f(x)”

Given the following Taylor series: $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}- \dots$ We know that: It converges for all of $x$ It converges to the function $\cos x$ The ...
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57 views

Approximating polynomial of higher degree

Suppose that we approximate a function $f(x)$ for $x$ near $0$ by a polynomial of degree $n$: $$f(x)\approx P_n(x)=C_0+C_1x+C_xx^2 + \dots + C_{n-1}x^{n-1} +C_nx^n$$ We need to find the values ...
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79 views

Finding a generating function for a pattern

I was working on this projecteuler.com problem, and I was very interested by the premise. Essentially, given n terms, find an ...
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48 views

Stuck with Taylor expansion of $f(x+x')$

I know that the Taylor series of $f(x)$ around $a$ is given by: $$f(x)=f(a)+f'(a)(x-a)+f''(a)\frac{(x-a)^2}{2}+\dots=\sum_{n=0}^\infty \frac{f^{(n)}(a) }{n!} (x-a)^n$$ In my textbook I see the ...
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52 views

Maclaurin series of: $ f(x) = {x + 5\over1-x^2}$.

I'm trying to get the Maclauren series of: $ f(x) = {x + 5\over1-x^2}$. I am sure there is some trick here, the result according to Mathematica is: $5 + x + 5x^2 + x^3 + 5x^4 + x^5 + 5x^6 + \ ...$ ...
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15 views

finding sequence for e converging at some speed

I want to find an infinite sequence that conerges to e so that the kth term of the sequence is less than 10^-k away from e. Obviously, I've considered the Taylor series, but asymptotic bounds on the ...
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46 views

Why does the moment of approximation matter for the end result?

I am trying to wrap my head around the reason why the moment of approximation matters for the end result of my analysis. As an example, let's take an equation for which we can still find the full ...
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96 views

Show the Newton method converges to 0 quadratically?

Using taylor series, show that if $x_n$ converges to a root, $f(x_n)$ usually converges to 0 quadratically. I reached a point I think I need to show that $\lim_{x\to \infty} ...
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146 views

What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
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172 views

Taylor series about different points implies different interval of convergence?

I'm considering the taylor series of functions whose radius of convergence is non-infinite about different points, and I'm not sure if I'm interpreting this correctly. Suppose, for concreteness, you ...
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255 views

Taylor polynomial about the origin

Find the 3rd degree Taylor polynomial about the origin of $$f(x,y)=\sin (x)\ln(1+y)$$ So I used this formula to calculate it ...
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37 views

Confusion related to Taylor series approximation

I found this Taylor series approximation given by $f(x_{\alpha}) = f(x) + \nabla f(x)'(x_{\alpha}-x) + o(||x_{\alpha}-x||)$. I didn't get how this $o(||x_{\alpha}-x||)$ term came from. Can anyone ...
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1answer
161 views

Explaining and using the $N$-term Taylor series for $\sin x$

So I'm given the Taylor Series expansion of the sine function and I've been asked to prove it (Done) and then construct the following by my lecturer: Explain why the Taylor series containing $N$ ...
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150 views

Maclaurin series for $e^z /\cos z$.

I want to find the Maclaurin series for the function $$f(z)=\frac{e^z}{\cos z}.$$ Right away I can tell that the radius of convergence will be $\pi/2$, since it's the distance to the nearest ...
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1answer
244 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?