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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5
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1answer
865 views

Taylor series for cosine around $\pi/3$

I need the Taylor-Series for $ f(x) = \cos(x) $ in $ a = \pi/3$: \begin{align*} f(x) &= \cos(x - \pi/3 + \pi/3) \\ &= \cos \left( x - \frac{\pi}{3}\right) \cos\left(\frac{\pi}{3}\right) - ...
-1
votes
2answers
82 views

maclaurin series for function undefined at a point

Say i want a power series for a function such as $$\frac{(2x+2)(x)}{(2x)(3x+1)}$$ at $x=0$. How would one go about this? I have acquired the second, third and fourth terms, but am struggling getting ...
2
votes
3answers
50 views

Expanding $(1 - x + 2y)^3$ in powers of $x-1$ and $y-2$ with a Taylor series

I would like to do this. I observe that I can write $$f(x,y) = (1 - x + 2y)^3 = (2(y-2) - (x-1) + 4)^3.$$ It's easy to do this via algebra directly. However, I'm asked to do it by computing the ...
0
votes
1answer
48 views

If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$

Prove: if $f''(x_0)$ exists then $\lim\limits_{x \rightarrow x_0} \dfrac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$. I'm not exactly sure how Taylor's theorem fits into all this, but I found ...
9
votes
1answer
315 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) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...
1
vote
1answer
28 views

Finding Taylor series of $x^{-3}$ about $x=a$

How to find the Taylor series of $x^{-3}$ about $x = a$? Usually I can do ones where $f(x) = (x+c)^{-3}$ but when $c=0$, I'm unsure. Even for positive exponents there's a simple way.
4
votes
1answer
107 views

Remainder in the multivariate Taylor expansion

For the function $f:\mathbb{R}\to\mathbb{R}$, I can write the Taylor expansion $$f(x+h) = f(x) + f'(x)h + \frac{1}{2!}f''(x)h^2 + O(h^3)$$ where the remainder is $o(h^2)$ as well. I'm confused with ...
8
votes
1answer
96 views

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 ?
1
vote
2answers
281 views

Proof that sum of power series equals exponential function?

I have found that the Sum series equal an exponential function as below, however I have not found a proof for it: $$ ze^z = \sum_{k=0}^{\infty} k \frac{z^k}{k!} $$ I have though managed to prove ...
1
vote
0answers
66 views

Taylor's Theorem for Multivariable Implict Functions

I'm trying to find the $2$nd order Taylor polynomial for $z=g(x,y)$ near the point $(\frac {\pi}{2}, 1,1)$, given the function $\sin(xyz)=z^2$. I've never found the Taylor polynomial of a function ...
2
votes
1answer
107 views

Bound remainder of Taylor series with Lipschitz property of derivative

I feel like this should be fairly simple: I would like to use the fact that $$|g'(x) - g'(y)|\leq C|x-y|^\delta$$ for all $x,y\in \mathbb{R}$ for some $C,\delta >0$ to put a bound on the ...
0
votes
1answer
45 views

Using Taylor series to create a zero function

Let $f(t)$ be a $n$th order polynomial with real, positive coefficients (I am not sure if these constraints are necessary). Then after taking $n+1$ or more derivatives, the function vanishes and is ...
3
votes
1answer
27 views

Equivalence of two expressions

I have the following expression $$\varphi = -5\Delta t + \sqrt{25(\Delta t)^{2}+1}$$ and I want to show that in fact this is equal to $$\varphi = e^{-5\Delta t} + \mathcal{O}(\Delta t^{3}).$$ To do so ...
3
votes
2answers
51 views

Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.

I was google-ing and came across this question. Till now I don't have any solution. Let $f$ be a double differentiable function on $(1,\infty)$. Let $M_0=\sup_{x\in R}|f(x)|$, $M_1=\sup_{x\in ...
1
vote
1answer
77 views

uniformly continuous when second derivative is bounded

Let $f$ be continuous on $[a,b]$. $f$ is twice differentiable on $(a,b)$ and $|f^{\prime\prime}(x)|\leq M$ for all $x\in [a,b]$. Show $f$ is uniformly continuous. This is a question from an exam in ...
3
votes
3answers
33 views

Finding the $n$th Taylor coefficient of $g(z)=\frac{z}{(z-b)^2}$ centered at $a$ (where $a=2-\sqrt{3}$ and $b=2+\sqrt{3}$?

I've introduced $a$ and $b$ in order to simplify the notation : $a=2-\sqrt{3}$ and $b=2+\sqrt{3}$. I'm trying to compute the Taylor Series for $g(z)=\frac{z}{(z-b)^2}$ centered at $a$. I denote the ...
2
votes
0answers
24 views

$f:[0,1] \to \mathbb{R}$ with $f(0) = 0 = f(1)$ and $|f''|\leq M$. Show $|f'(1/2)| \leq \frac{M}{4}$. [duplicate]

I attempted a solution using this version of Taylor's Theorem, \begin{align*} |f(d) - P_n(d)| &= \left| \frac{f^{n+1}(t)}{(n+1)!} \right| |d-c|^{n+1} \\ |f(d) - f(c) - f'(c)(d-c)| &= \left| ...
3
votes
1answer
46 views

Find a polynomial which approximates $f(x) = \sqrt{x}$ in the interval $(4,5)$ within $10^{-8}$

Find a polynomial which approximates $f(x) = \sqrt{x}$ in the interval $(4,5)$ within $10^{-8}$. I've been trying to solve this using Taylor's remainder theorem, $$ |f(d) - P_n(d)| = \left| ...
3
votes
2answers
125 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 ...
1
vote
1answer
37 views

Taylor series of an implicit function

Suppose the function $s:[-\delta, \delta] \to \mathbb{R}$, $\delta > 0$, is defined implicitly by $$s(t) = 1 - c\beta t (s(t))^{\beta}$$ for some $c > 0$, $0 <\beta < 1$. Can an ...
5
votes
0answers
67 views

Manipulation of Taylor/Laurent series

I have a question regarding how to expand a given rational function into its Taylor/Laurent series representation. Suppose we are given the function $$f(z) = \frac{z}{(z-1)(z-3)},$$ and are asked to ...
2
votes
1answer
64 views

Evaluate Infinite Sums

I'm having trouble with this question. $$E=\sum_0^\infty \frac{nhve^{-nx}}{\sum_0^\infty e^{-nx}}$$ where x=(hv)/(kT) Evaluate both sums and show that E=hv(e^x -1)^-1 I've tried comparing ...
1
vote
1answer
34 views

How do I perform taylor expansion of the following?

Taylor expansion about $(x,y)$ of $f(x + a,\; y + k\; f(x + b,\; y + c))$ I do not understand what happens to the second $f$ inside. The inspiration for this question is Runge-Kutta methods.
3
votes
2answers
80 views

Showing that the integral remainder of the Taylor expansion of $f(x)=-\log(1-x)$ goes to $0$

Let $|x|<1$. Define $R_n(x):=\int_{0}^{x}\frac{(x-t)^{n-1}}{(1-t)^n}dt$. How do we prove that $\lim_{n\to \infty}R_n(x)=0$? This is actually the integral remainder of the Taylor expansion of the ...
1
vote
1answer
41 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 ...
1
vote
1answer
37 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 ...
1
vote
0answers
43 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 ...
0
votes
1answer
39 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 ...
2
votes
1answer
44 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 ...
2
votes
4answers
246 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 ...
4
votes
0answers
250 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} ...
0
votes
0answers
41 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 ...
2
votes
2answers
103 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 ...
2
votes
2answers
51 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) = ...
0
votes
2answers
37 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 ...
5
votes
2answers
173 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). ...
2
votes
3answers
110 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 ...
2
votes
1answer
67 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 ...
1
vote
1answer
39 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. ...
0
votes
1answer
23 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 ...
2
votes
1answer
108 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. ...
0
votes
1answer
21 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: ...
2
votes
0answers
40 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 ...
0
votes
2answers
23 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 ...
2
votes
1answer
16 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 ...
1
vote
1answer
45 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 ...
1
vote
0answers
27 views

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 ...
3
votes
2answers
130 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 ...
1
vote
1answer
73 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 ...
0
votes
2answers
102 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 $f(x) \leq Cg(x)$ ...