Tagged Questions

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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0
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1answer
17 views

what's the significance for adding upto n derivatives of a function in taylor series [on hold]

Why do we say that upto n derivatives of a function are added as in taylor series to get a function? $f(x)$=$f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots$ Is there anything that slope ...
0
votes
1answer
9 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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0answers
17 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 ...
2
votes
1answer
92 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
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1answer
11 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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0answers
6 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 ...
2
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0answers
28 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{(k-1)A_{k-1}(t)-kA_k(t)}{\alpha+2\beta t} + \delta_{k \beta} . ...
2
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0answers
28 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 ...
7
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1answer
51 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 ?
3
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0answers
83 views
+150

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
16 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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1answer
17 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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0answers
46 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 ...
2
votes
1answer
11 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
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1answer
24 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
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0answers
20 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 ...
-1
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1answer
18 views

Taylor polynomial Two variables [closed]

Find the taylor polynomial of order 4 of f(x,y)=1/(1+3xy) based at the point (0,0)
1
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1answer
31 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
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1answer
41 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: ...
0
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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 ...
0
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2answers
47 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 $f(x) \leq Cg(x)$ ...
3
votes
2answers
36 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 ...
0
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0answers
18 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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2answers
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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0answers
18 views

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 ...
1
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0answers
25 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 ...
0
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0answers
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 ...
2
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0answers
8 views

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 ...
2
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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 ...
0
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0answers
14 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 ...
1
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0answers
23 views

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 ...
1
vote
5answers
203 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 ...
0
votes
1answer
14 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 $cos(1/z)$ ...
0
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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) = ...
2
votes
3answers
43 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 ...
1
vote
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 + ...
7
votes
1answer
277 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
24 views

Finding a Taylor Series representation of $f(x)=\ln(\frac{1+2x}{1-2x})$ centered at $0$.

I'm trying to find a Taylor Series representation of $f(x)=\ln(\frac{1+2x}{1-2x})$ centered at $0$. So I am using the Maclaurin Series representation of $f(x)=\ln(1+x)$ which is ...
3
votes
5answers
98 views

When $a\to \infty$, $\sqrt{a^2+4}$ behaves as $a+\frac{2}{a}$?

What does it mean that $\,\,f(a)=\sqrt{a^2+4}\,\,$ behaves as $\,a+\dfrac{2}{a},\,$ as $a\to \infty$? How can this be justified? Thanks.
0
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1answer
24 views

Multiple representations of ternary expansions of numbers

$x \in [0,1]$. If in binary expansions ie series $\displaystyle x = \sum_{i=1}^{\infty} \frac{x_i}{2^i}$ where each $x_i \in \{0,1\}$ we identify the sequences $\underline{x}$ and $\underline{x}'$ ...
0
votes
1answer
11 views

Use the power series representations of functions to find the taylor series of $\frac{1}{5+x'}$ at center = -6.

I am trying to find the taylor series of $f(x)=$ $\frac{1}{5+x'}$. And I cannot seem to get how to find the taylor series using the method I've been using for other functions. Another thing that's ...
0
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1answer
25 views

MacLaurin of the Third-degree in sin(a*x)*cos(b*x) at given values

Alright so from my understanding MacLaurin is a special case of Taylor Series but at f(0). However my question involves solving the third degree of MacLaurin for $$f(x) = sin(a \times x)\times ...
1
vote
2answers
27 views

Sum Representation of log(1 + x)

$\log(1+x) = \sum_{k=1}^{\infty} \left(\dfrac{x}{1+x}\right)^{k} \dfrac{1}{k} = \sum_{k=1}^{\infty} \left(1 - \dfrac{1}{1+x}\right)^k \dfrac{1}{k}$ Why is this true? The most sum representation of ...
1
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4answers
227 views

Maclaurin series expansion of $\frac{1}{(1+x)^n}$

I am trying to figure out the Maclaurin Series expansion of the function, preferribly in a sneaky and clever way. Any ideas? Thanks.
1
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1answer
64 views

Taylor Polynomial of $f(x)=\cos(x)\cdot\sin(x)$

How would I calculate the third maclaurin/taylor polynomial on $\cos(a) \cdot \sin(b)$, Do I use the product rule when I calculate the derivatives? I don't know where to start or read about it, been ...
0
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1answer
20 views

Expand with a Taylor formula $\frac{2+x}{x^2+2x+2}$ near the $x_0 = -1$

I am not sure whether I am doing it correctly. So, $$\frac{2+x}{x^2+2x+2} = \frac{2}{x^2+2x+2} + \frac{x}{x^2+2x+2} = F_1 + F_2,$$ $$x^2+2x+2 = (x - x_1)(x-x_2), \text{where} \\ x_1 = i+1, \\x_2 = i ...
0
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0answers
18 views

The remainder of a Taylor Polynomial.

I am looking at a problem with Taylor series, and I'd just like to know if I am doing it correctly, or at least headed in the right direction. I start by finding the Taylor series for $arcsin(x)$ ...
0
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0answers
13 views

Understanding this question in regards to Taylor Polynomials

I am looking at the following question, and I simply don't understand it. I've calculated the Taylor series for $f(x) = \arcsin(x)$ centered around $0$ of order $n=3$. Evaluating this series at $x = ...
1
vote
2answers
25 views

Simplification of a series so that it converges to a given function

I am trying to rearrange the series $ \frac{1}{1-z} - \frac{(1-a)z}{(1-z)^2} + \frac{(1-a)^2z^2}{(1-z)^3} - \cdots$ In such a way that I can show it converges to $\frac{1}{1-az} $ What I ...
1
vote
1answer
36 views

The $n$th-derivative of $q(x) = x^4 - 8x^3 - 4x^2 + 3x - 2$, where $n \le 4$

Some factors will be $\frac{4!}{(4-n)!}\cdot a_nx^n, \frac{3}{(3-n)!} \cdot a_{n-1}x^{n-1}, \ldots, \frac{1!}{(1-n)!} \cdot a_0x^0$, but the lowest degree one will always become zero in the next ...