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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9
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1answer
312 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
36 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
102 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
votes
1answer
38 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
19 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
votes
1answer
36 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
32 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
vote
4answers
253 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
vote
1answer
66 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
votes
1answer
21 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
votes
0answers
36 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 $arcsin(x)$ ...
1
vote
2answers
27 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
39 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 ...
0
votes
0answers
37 views

Why is it true that $\forall b\in(0,1): (1-b)\left(e(1-b)\right)^{\frac{b}{1-b}}\geq\prod\limits_{n=2}^{\infty}n^{-b^n}\geq 0$

Why is it true that $$\forall b\in(0,1)$$ $$1\geq(1-b)\left(e(1-b)\right)^{\frac{b}{1-b}}\geq\prod\limits_{n=2}^{\infty}n^{-b^n}\geq 0$$ Note: Let $$f(x)=\prod\limits_{n=2}^{\infty}n^{-b^n}$$ Then ...
0
votes
1answer
55 views

Help finding n-order Maclaurin polynomial

EDIT AND PLEASE NOTE: I DON'T want solutions that are nicer or more elegant but presume knowledge of other infinite series and/or don't come from the nth-derivative because I'm precisely studying how ...
1
vote
1answer
19 views

Can anyone explain how to show the finite difference equation $y'_{0}=\frac{y_{1}-y_{-1}}{2h} + O(h^{2}).$?

I was given that $y_{j}=y(x_{j})$ where $x_{j}=x_{0}+jh$ for integer j and positive h. I need to show that $$y'_{0}=\frac{y_{1}-y_{-1}}{2h} + O(h^{2}).$$ I thought I could start by finding the Taylor ...
3
votes
1answer
32 views

Are the Taylor polynomials of a function the results of a minimization problem?

Here is an example to better explain my question. Consider the function $f(x) = \cos(x)$. I want to approximate it in the set $[-\pi; \pi]$ using a polynomial $g(x) = a + bx + cx^2$ of order $2$. ...
2
votes
0answers
62 views

Book Request: Taylor's Theorem for functions $f: \Bbb R^n \to \Bbb R^m$

I'm looking for a resource (e.g. a book, website, or arxiv paper) that goes over the general case of Taylor's theorem, with a full proof and examples. Do you guys know of any material that covers ...
2
votes
3answers
58 views

Why is this true? $\forall a\in(1,\infty), B\in(0,\infty), x\in(0,\infty) : a^x\geq \left(\frac{ex\ln(a)}{B}\right)^{B}$

I know $$\forall a\in(1,\infty), B\in(0,\infty), x\in(0,\infty)$$ $$a^x\geq \left(\frac{ex\ln(a)}{B}\right)^{B}$$ can be proved using AM-GM. Is there a simple way to show the inequality holds in all ...
2
votes
2answers
38 views

Approximation of a ratio

Is this approximation true? If so, why? $$\frac{1+x}{1+y}\approx 1+x -y$$ I think it has something to do with $x$ and $y$ being close to zero, so that the ratio of the two is approximately equal to ...
1
vote
1answer
73 views

Laplace equation, Taylor expansion

I couldn't find it anywhere, so I decided to write my question here: I have problems solving this equation: $$u_{xx} + u_{yy} = 4,$$ subjected to the conditions $$u(x,x)=2x^2, \quad u_x(x,x)=2x$$ ...
0
votes
1answer
46 views

The accuracy of approximating $ f(x) = x^{2/5}$ for $0.9 \le x \le 1.1$ using the cubic Taylor polynomial

For the equation $ f(x) = x^{2/5}$, $a=1$, $n=3$, $0.9 \le x \le 1.1$ I was able to approximate f by the following Taylor polynomial: $$ F_3(x) = 1 + \frac2 5 (x-1) - \frac3{25}(x-1)^2 + ...
1
vote
2answers
257 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
13 views

Symmetry of convergence around a of a Taylor Polynomial

I heard the lecturer mention this shortly, so I'd like a more detailed explanation of it. He showed us a taylor approx. of ln(x) at x = 1, and showed that it only matches in the interval ]0,2], so ...
2
votes
2answers
28 views

Taylor series of polynomial.

I know that the taylor approx. of a polynomial centered at 0, if n gets big enough, is just the polynomial itself. But why do people always say "centered at 0"... wouldn't we also get the polynomial ...
0
votes
1answer
36 views

Maclaurin series accuracy

Find an $n_1$ such that the $n_1$th-order Taylor polynomial for $\sin(x)$ about $x=0$ gives an approximation of $\sin(x)$ with an error of less than $5\cdot 10^{-10}$, for all $x$ between $0$ and ...
0
votes
1answer
13 views

Bounding $|n^2(1-(\cos\frac{1}{n})^2)|$

I'm working on a a problem that involves me needing to give an upper bound for the following expression: $|n^2(1-(\cos\frac{1}{n})^2)|$ My attempts at bounding it: Expanding the expression: ...
0
votes
1answer
59 views

On the multivariable Taylor expansion

Apparently the second order multivariable Taylor expansion is: $$f(\mathbf x+\mathbf h)=f(\mathbf x)+ \partial_i f(\mathbf x) h_i + \frac 12 \partial_j \partial_i f(\mathbf x + t \mathbf h) h_i h_j$$ ...
1
vote
2answers
75 views

Taylor series of Infinitely differentiable function with nonnegative derivatives

Let $f(x)$ be a nonnegative and infinitely differentiable function on $[-a,a]$ to $\mathbb{R}$ such that $\forall x\in[-a,a]:f^{(n)}(x)\ge0$. Prove that the series: $$\sum_{i=1}^\infty ...
0
votes
2answers
43 views

Proof of an inequality using Newton's Method

Question: Show that the function $f(x):= x^3 -2x -5$ has a zero $r$ in the interval $I:= [2,2.2]$. If $x_1 :=2$ and if we define the sequence $(x_n)$ using Newton's procedure, show that $|x_{n+1} -r| ...
1
vote
1answer
61 views

Is there a difference between $a \cdot a^T$ and $a^2$?

The title says it all... I can't see the difference between $a \cdot a^T$ and $a^2$, when $a$ is a vector. However I encountered a formula stating $$\frac{1}{|y+a|} = \frac{1}{|y|} - \frac{y \cdot a ...
1
vote
0answers
42 views

Functions f(x) equal to Taylor series vs Fourier series vs Bessel series

(I had trouble phrasing the question below due partially to the fact that Bessel functions $J_{\alpha}(x)$ and $U_{\alpha}(x)$ are defined for any complex $\alpha$, so below I tried to express an ...
0
votes
3answers
57 views

If subsequent terms keep getting larger, does that mean no limit exists?

Take the following Taylor expansion: $$ \dfrac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots $$ This only holds for $ 0 \leq x < 1. $ Let's say you want to prove this doesn't hold for $x>1$. You can ...
0
votes
1answer
46 views

Determining the domain of holomorphic function, the taylor series of function with its convergence's radius.

I need some help and correct my knowledge, please. Let $f(z)=(e^{z}-1)/(1+z+z^{2})$. Determine the largest domain $\Omega$ in $\mathbb{C}$ such that $f$ is holomorphic in $\Omega$. Since ...
0
votes
0answers
28 views

Taylor expansion of a scalar function

I have an expression on the form $$ \sum_{i=1}^N{\rho_i}f(\mathbf x+\mathbf c_i)\mathbf c_i $$ where $\rho_i$ is a scalar, $f(\mathbf x+\mathbf c_i)$ a scalar function of the vector quantities ...
0
votes
1answer
41 views

Real Analysis: Taylor's Theorem Approximation Proof

If $x>0$ Show that $\lvert (1+x)^{(1/3)} - (1+\frac{x}{3} -\frac{x^2}{9}) \rvert \le (\frac {5}{81})x^3$. Use this inequality to approximate $1.2^{1/3}$ & $2^{1/3}$. That is the actual ...
3
votes
1answer
59 views

manipulations with Taylor expansions for log and sinh

How could we derive the equality $$ \frac14 \sum_{m=1}^\infty \frac1m \frac{1}{\sinh^2 \frac{m\alpha}2} = - \sum_{n=1}^\infty n\log (1-q^n)$$ where $q=e^{-\alpha}$ ?
0
votes
1answer
67 views

Constructing a sequence of function with bounded derivative

Let $f:\mathbb R\mapsto\mathbb R$ be a smooth function and analytic at $x=0$. I wish to find a sequence of functions $\{f_n\}$ such that $\{f_n(x)\}$ is convergent to $f(x)$ for all $x$ and $f'''_n$ ...
6
votes
0answers
127 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider ...
1
vote
2answers
37 views

Series Coefficient Convergence implies Uniform Convergence

Trying to find a reference for the following. Define the entire functions, $$f_n(x)=\sum_{k=0}^\infty a_{n,k}x^k\ \ \ \ \ \ \ \ \ \ \ f(x)=\sum_{k=0}^\infty a_kx^k.$$ If for each $k$, ...
1
vote
6answers
40 views

Taylor expansion square

Consider the following expansion $$\sqrt{1+x} = 1 + \dfrac{1}{2}x - \dfrac{1}{8}x^2 + \dfrac{1}{16}x^3 .. $$ Show this equation holds by squaring both sides and comparing terms up to $x^3$. I ...
0
votes
1answer
37 views

Prove $\frac{dy}{dx}$ is approximated by $\frac{y(x+h)-y(x-h)}{2h}$ to $O(h^2)$

I tried to solve it by truncating the Taylor series expansions for $y(x+h)$ and $y(x-h)$ but I couldn't find a way to relate it to the derivative. I wasn't sure where the appropriate place to truncate ...
0
votes
2answers
68 views

Why cant we do substitution in differentiation but is ok in taylor series?

I have the same question 10 year ago when i was studying high school. I dont understand it and i give up the math. 10 year ago, i need to work with calculus during work and this question come to find ...
2
votes
2answers
70 views

Calculators using Taylor polynomials?

I've always heard that calculators (TI-84's and the like) use Taylor polynomials to approximate trigonometric/exponential/etc functions. Do any of you know this for a fact?
0
votes
1answer
28 views

error order evaluation in taylor expansion of a definite integral

I have a function $g(x)=f(x)e^{-x}$ and i want to consider the following integral: $\int_{0}^{\infty}g(x)dx$. Since $f(x)$ is a complicated, but monotonic decreasing, function in the interval ...
0
votes
1answer
59 views

Prove error bound using Taylor's series Error term (Bound doesn't seem to make sense)

I have to prove that at least seven terms must be used in the Taylor series estimation of x - sin(x) in order for the error to be <= $10^{-9}$. This doesn't seem correct however. This series is ...
0
votes
1answer
29 views

I do not know the point at which this Taylor series was derived, can someone explain please?

I am required to derive Euler's method through Taylor's Theorem. I have been given the Taylor series for $y(t)$ as shown below. However I do not understand what point the Taylor series was derived. ...
1
vote
1answer
21 views

Error estimate of definite integral of a taylor expanded function

If I consider a monotonic decreasing function $f(x)$ in the interval $[0,+\infty[$, and I consider the definite integral $\int_{0}^{+\infty}f(x)\,\mathrm{d}x$. What is the error committed if I compute ...
3
votes
2answers
46 views

Other ways to evaluate $\lim_{x \to 0} \frac 1x \left [ \sqrt[3]{\frac{1 - \sqrt{1 - x}}{\sqrt{1 + x} - 1}} - 1\right ]$?

Using the facts that: $$\begin{align} \sqrt{1 + x} &= 1 + x/2 - x^2/8 + \mathcal{o}(x^2)\\ \sqrt{1 - x} &= 1 - x/2 - x^2/8 + \mathcal{o}(x^2)\\ \sqrt[3]{1 + x} &= 1 + x/3 + \mathcal{o}(x) ...
0
votes
1answer
51 views

approximation of x=sin(x) error

The larger $x$ becomes, the worst the approximation of $x=\sin(x)$ becomes. How large can $x$ get before the error is no longer less than $1/(2\times10^{14})$? What I have tried: I know that the ...