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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23 views

How correctly apply a Taylor expansion of first order to a multivariate function decomposition.

Suppose I have two function $f(u,v):\mathbb{R}^2\to\mathbb{R}$ and $g(r,s):\mathbb{R}^2\to\mathbb{R}$. I'm interested in the first order Taylor expansion of $$h:=f(u,g(r,s))$$ To be precise, for a ...
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1answer
35 views

Find a formula for Taylor series of $\left(\frac{1}{1+z^2}\right)^n$

So the way I think I should approach this is by getting a result for $n=1,2,3...$ and then examine them. I could easily get the Taylor series expansion for $n=1$, but then I don't really know how to ...
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0answers
31 views

Statement regarding Taylor's theorem

I have this statement on Taylor's Theorem in notes : Let $U$ be an open set in $\mathbb R^n$ . Let $f\in C^{m+1}(U,\mathbb R)$.Let $x\in U$ and choose $r\gt 0$ such that $B(x,r) \subseteq U$.Then ...
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2answers
57 views

Analogy to the purpose of Taylor series

I want to know an analogy to the purpose of Taylor series. I did a google search for web and videos : all talks about what Taylor series and examples of it. But no analogies. I am not a math geek and ...
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1answer
30 views

Showing equality of taylor series

Use the taylor series for $\frac{1}{\sqrt{1-x}}$ to show that the sum from n = 0 to infinity of $\frac{1}{8^n} {2n\choose n} = \sqrt2$ I have the taylor series as the sum from n = 0 to infinity of ...
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1answer
40 views

Importance of Taylor polynomials

I am reading a book and it says that if $T_n(x)$ is the Taylor polynomial of $f$ of order $n$ at $x=a$ then $\lim_{x\rightarrow a} \frac{T_n(x)-f(x)}{(x-a)^n} = 0$. In other words, the error is ...
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4answers
30 views

What does it mean intuitively for a Taylor Series to be centered at a specific point?

I understand what a Taylor series is and how to find the Taylor series of a function. However I do not understand intuitively what it means to find a Taylor series for a specific function, centered at ...
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16 views

Compute the guaranteed error term as given by Taylor's theorem

This is an exercise from Calculus-2 course: Find the first three nonzero terms in the Taylor series for $\tan x$ on $[-\pi/4,\pi/4]$, and compute the guaranteed error term as given by Taylor's ...
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24 views

A calculus-2 exercise related to Taylor's theorem

This is an exercise of Taylor's Theorem topic. How many terms of the series for $\log x$ centered at 1 are required so that the guaranteed error on $[1/2,3/2]$ is at most $10^{-3}$? What if the ...
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12 views

Order notation problem

I have got stucked for order notation and expansion. Can you give me some clear hint?
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2answers
67 views

Taylor series expansion for $e^{-x}$

could anyone show me the Taylor series expansion for $e^{-x}$.I was trying to find out how $e^{-i\theta}$=$\cos\theta-i\sin\theta$. More specifically could you show me how ...
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2answers
49 views

What would be a power series for $f(z)=\sin(z)$ centered at $1$?

Everything is in the question! I've seen loads examples like "centered at $\pi$, $\pi/2$,... But $1$ would make everything much different... I've tried to work this way: $\sin(z) = \sin((z-1)+1) = ...
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3answers
59 views

$f: \mathbb R \to [0, \infty ) $ be a twice differentiable function with $f'' \le 0 $ , then how to show that $f$ is constant ?

Let $f: \mathbb R \to [0, \infty ) $ be a twice differentiable function with $f''(x) \le 0 , \forall x\in \mathbb R $ , then how to show that $f$ is constant ? My work:- Consider arbitrary $x\in ...
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1answer
48 views

How to approximate this nasty exponential function with an integral?

What is the best way to approximate a function of the following form, $$ \text{exp}\left(-\int_{y}^{+\infty} f(x)\ dx \right)$$ Any approximation to this, does taylor series work? The reason I am ...
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1answer
26 views

Is there any realtionship between linear approximation and taylor series?

As said at Where did the linear approximation/linearization formula come from? about linear approximation is there any thing that relates taylor series and linear approximation. ...
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1answer
11 views

Use Taylor's method to determine the constants a and b in the 4th order Adams-Bashforth method.

Use Taylor's method to determine the constants a and b in the 4th order Adams-Bashforth method; $$w_{i+1}=w_i + \frac{h}{24}(55f_i +af_{i-1} + 37f_{i-2} + bf_{i-3}).$$ I am not sure how to begin ...
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23 views

Taylor polynomial for $\sin(2x)$ about $x=(\pi/2)$, order $2n-1$.

In my assignment I have to find the Taylor polynomial for $\sin(2x)$ about $x=(\pi/2)$, order $2n-1$. And I just made $P_4(x)$ to figure out ...
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1answer
95 views

Prove this formula for $\pi$

I have to use a certain approximation for $\pi$ for my computer science class, but I don't really understand what's going on, other than that this is related to the Taylor polynomial for arctangent. ...
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1answer
12 views

Functions of Complex Variables - Find the first 4 terms of the Taylor Series.

I have been asked the following question: Find the first four terms of the Taylor Series of the following function about 0. $$ f(z)=\frac{e^z}{(1+z)} $$ I know that the solution to this question is: ...
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8 views

Differentiational equation construct power series expansion

I got a question In order to improve the accuracy of your numerical estimate you are to use a power series expansion of y(x)to estimate y(1). (You may find it easier if you multiply both sides of ...
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41 views

Validity of approximating a difference equation with a differential equation

Consider the following difference-differential equation defined for positive integer indices $k$ and $t$: $$ A_k(t+1)-A_k(t)=\beta \frac{ A_{k-1}(t)- A_k(t)}{\alpha+2 t} + \delta_{k \beta} . \tag{1} ...
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30 views

Taylor polynomial converging pointwise but not uniformly?

Many standard examples of Taylor series $(\exp(x), \sin(x), \cos(x))$ converge uniformly, others don't converge to its original function at all, e.g. $\exp(-x^{-2})$. I couldn't think of any smooth ...
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Newton's way of getting a Taylor expansion

I don't understand how Newton find the Taylor expansion of $\frac{a^2}{b+x}$ by the following method : **This screenshot is from : The method of fluxions and infinite series Any idea ?
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2answers
145 views

Closed formula for the asymptotic limit of a definite integral

I would like to solve the following integral: $$ I_0 (a,b)= \int_0^1 dx\int_0^{1-x} dz \frac{1}{a z (z-1)+a x z + x(1-b)}$$ in the limit where $b$ is small ($a$ and $b$ are positive constants). ...
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2answers
18 views

Series expansion around natural logarithm

I am working on an integral using the Laplace-method, and I have to do a series expansion of the following $$ \phi(x,t) = x \ln(t) - t, $$ according to the solution the answer is $$ \phi(x,t) = x ...
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33 views

Obtaining the Taylor Series Method, order two

So here's the problem I'm having trouble with... The initial value problem: $$ x'(t) = \cos(x+t^2) $$ with $x(-5) = -1$ on the interval $[-5, 3]$. I have no idea how to star this problem off ...
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55 views

”Mehrstellenverfahren” of Collatz

I can do Taylor expansion on the Left hand side but I would like to know how to do Taylor expansion on the right side. Can anyone help me with finding the Taylor expansion for the double deivatives ...
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1answer
12 views

Series Expansion

Find the series expansion of ln(1+e^(-z)) when z is very large. I figured this out for when z is very small but I am unsure what to do if z is very large. I think if I just get a hint about that I ...
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1answer
28 views

Can we do Taylor approximation in one direction

Let $f:\mathbb{R}^2\to\mathbb{R}$. Can we do Taylor approximation for only one variable $$f(x,y) \approx f(x_0,y) + \frac{\partial }{\partial x}f(x_0,y)(x-x_0) + \frac{1}{2}\frac{\partial^2}{\partial ...
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Estimate error on square root simplification

I have following term: $\sqrt{(\gamma+2)^2+4\gamma}$. I know that I could be able to simplify it to: $\sqrt{(\gamma+2)^2+4\gamma}$ $\approx$ $(\gamma + 2) + 2\sqrt{\gamma} + \epsilon$ and that this ...
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1answer
34 views

Find the first three terms of the maclaurin series of $\tanh(z)$ and its radius of convergence

This is my first time dealing with maclaurin series of complex variables. Here is my attempt: Since $\tanh = \frac{\sinh(z)}{\cosh(z)}$, the maclaurin series is valid when ...
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1answer
45 views

Arctan(f(x)) is almost the same as Erf(f(x)) for many f(x). Is the just coincidence or is there a reason?

For example: Arctan(x) is almost Erf(x) (subtle differences in absolute value and curve) Arctan(x^50) is almost Erf(x^50) (difference in absolute value) and many others, so we can conclude: ...
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1answer
21 views

Find the series expansion of $\left(e^{(x-1)}\right)^2$

Find the series expansion of $\left(e^{(x-1)}\right)^2$. I thought maybe I could use binomial expansion but that is only for $(1+x)^n$, so now I am unsure how to proceed. I could set $(x-1)^2=n$ and ...
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2answers
51 views

What does the constant mean in Big O notation?

I have a big issue in understanding the real meaning of Big O notation. Classical definition: $f(x) = O(g(x))$ as $x\rightarrow k$ if there exist $\delta, C > 0$ such that $f(x) \leq Cg(x)$ ...
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2answers
38 views

Find series expansion of 1/cosx

Find the series expansion of 1/cosx from basic series expansions. I tried to find 1/cosx from the expansion of cosx but was unsure how to continue. When I found 1/cosx from the basic formula for ...
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22 views

Find the minimum number of terms in the series it would take before the error would be guaranteed to be less than 10^-9

Consider the function $f(t)=\ln t$ about the point $t_0=1$. Find the minimum number of terms in the series it would take before the error would be guaranteed to be less than $10^{-9}$.
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21 views

Is there a mistake in this page on asymptotic expansions?

I think there is an error in section 4.3 of this page - http://aofa.cs.princeton.edu/40asymptotic/ The author says that by taking $x = -\frac{1}{N}$ in the geometric series $\frac{1}{1-x} = 1 + x + ...
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How do I solve for the elements of the partial derivative of a Hessian matrix?

In the paper about Speeded-Up Robust Features, it says that in order to localize points, interpolation of nearby data is needed to find the location in space and scale. This is done by fitting a 3D ...
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26 views

Will this method find the taylor expansion of ANY function $f(x)$?

Polynomials are themselves Taylor expansions, correct? ex. $4x+5x^2+3 = 3+4x+5x^2 +0x^3 +0x^4 + \dots$ I'm assuming has no closed form besides $\sum_{n=0}^{2}(3+n)x^n + \sum_{n=3}^{\infty}0x^n$ but ...
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20 views

Determine f(z) by evaluating the sum

Determine an explicit expression for $f(z)$ by determining the sum of the series $f(z) = \sum_{n = 1}^\infty \frac{1}{n}$ $\cdot (\frac{z}{z-1})^n$ where $z\ne 1$ Yeah... I really don't know where ...
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Upper bound on derivatives of very high order?

I am doing a calculation where I am estimating a value $\omega$ by a Taylor polynomial. I know that $\omega \cdot a = f(b)$ and thus I can estimate $\omega$ by $a \cdot T_n f(b) $ where $T_n$ is the ...
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5answers
54 views

Prove that $e^x \ge$ its Maclaurin polynomial with n terms [closed]

a) show that $e^x \geq 1+x$ for all $x\geq 0$ b) deduce that $e^x \geq 1+x+\frac{1}{2}x^2$ for $x\geq0$ c) use induction to prove that for $x\geq 0, n\in \mathbb{N}$ $$e^x\ge ...
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16 views

The error function in the taylor's theorem for taylor series

I was reading taylor's theorem at wikipedia and at some point they say that $f(x)$ can be written as a function related to its linear approximation $P_1 = f(a) + f'(a)(x-a)$. This is a very simple ...
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If I have an infinite series, how do I know that the digits I calculated are rigth?

For example, there are infinite series for $\pi$, $e$, $\phi$... But if I sum a finite ammount of terms, I get an approximation for the series. How do I know how much correct digits of this ...
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5answers
212 views

Why are Maclaurin series useful if we can only use them for such a small range of numbers?

Okay, I am beginning to get how Maclaurin series work, but what I don't understand is why they are useful. Why would you want an infinite expansion for a series that works for such few values (only ...
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1answer
16 views

Singularity and residue in z = 0

How can I classify the singularity in $z = 0$ and determine the respective residue in $z = 0$ for the following function ? f(z) = $ cos(1/z)(z+1)^2$ Do I have to use Taylor expansion of $cos(1/z)$ ...
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2answers
35 views

Cannot expand $\sin(2x^2-4x+3)$ at $x_0 = 1$

Trying to expand $\sin(2x^2 - 4x+3)$ at $x_0 = 1$ to the $O(x-x_0)^n$. After substitution $t = x - 1 $, the problem becames $$\sin(2t^2+1) \text{ at } t_0 = 0$$ While we know that $$\sin(s) = ...
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3answers
46 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 ...
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2answers
24 views

Importance of the first term in a Taylor series

Suppose you have a function $f(x)$ whose Taylor series can be represented as the power series $$a_0 + a_1x^2+a_2x^4+...$$ If you are told that for $x\in\mathbb{R}_+$, $$a_0 + a_1x^2 + a_2x^4 + ...
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1answer
304 views

Infinite Series -: $\psi(s)=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.

We have a given converging series using derivatives and matrices(Analogue to Taylor's series) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...