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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4
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1answer
56 views

Odd nature of sine function. (Taylor series)

Although, this might be silly question. I am just wondering what happens to the odd nature of $\sin \theta$ when I expand it about some $ \pi/4 $. There are terms with even powers appearing as well. ...
2
votes
1answer
114 views

Taylor (Maclaurin) Series remainder for ${\rm sin}\ (x)$

So I just finished doing this problem and I think the solution I got is wrong, it seems a bit too large. According to my calculations, I need 36 terms. I fear I've made a mistake and I would really ...
3
votes
4answers
70 views

Taylor series of $(1+x)\ln(1+x)$ in $x=0$

How to determine the Taylor series of $(1+x)\ln(1+x)$ in $x=0$? My idea is finding the second derivative of the expression, which is $\frac{1}{1+x}$. The Taylor series of this expression is ...
1
vote
1answer
71 views

Maclaurin Series of $\int_0^x \cos t^2\,dt$

Find the Maclaurin Series for $\int_{0}^{x}\cos t^2\,dt$. $$\cos(x) = \sum\frac{(-1)^n x^{2n}}{2n!}$$ I'm trying this: $$\cos^2 x = \sum\frac{(-1)^n x^{4n}}{(2n!)^2}$$ How would you solve this ...
8
votes
3answers
424 views

Evaluating $\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$

How can we obtain following formula? $$\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}=\frac{1}{3}e^{\frac{-x}{2}}x\left(e^{\frac{3x}{2}}-2\sin\left(\frac{\pi+3\sqrt{3}x}{6}\right)\right).$$ I think if we ...
1
vote
3answers
397 views

First $3$ non-zero terms of the Maclaurin Series $\frac{1}{\sqrt{4+x^3}}$

Since each derivative will be multiplied by $3x^2$, are all the terms of this Maclaurin series $0$?
3
votes
2answers
132 views

Please help me understand Rudin Theorem 5.15

I am having trouble understanding the intuition behind the last part of this theorem. I'd appreciate some help understanding the intuition behind the last equation: $f(\beta ) = P (\beta ) + ...
1
vote
2answers
51 views

Estimating error in Taylor polynomial

Consider the nth order Taylor polynomial for cos x centered at 0 dented T(n) (x,0). How larger must we take n to guarantee that the error |cos x-T(n) (x,0) |is at most 10^-3 for x in [-pi/2,pi/2)
1
vote
2answers
90 views

question about taylor series

Can someone explain why 1 and 2 use different Taylor series? Why i cant use $1/(1+r)$ = $\sum_{n=0}^{inf}(-1)^n r^n$ on 2,vice versa?
2
votes
1answer
68 views

Linearize a simple ODE

This is homework. I have $\displaystyle \qquad S\frac{dh(t)}{dt} + \frac{1}{R}\sqrt{h(t)} = q(t)$ and need to linearize it. Setting all derivatives to zero, I get the steady-staty value of $h - ...
0
votes
2answers
55 views

What is the characteristic function used for?

Im totally new to statistics , but what is the characteristic function for ? I do not get that. I was reading about the bell curve and the Central Limit Theorem , but I did not get what the ...
2
votes
1answer
122 views

Laurent series, radii of convergence.

I'm working on the following exercise: Prove that a Laurent series \begin{align*} \sum_{n = -\infty}^\infty a_n(z-z_0)^n = \sum_{n = 0}^\infty a_n(z-z_0)^n + \sum_{n = 1}^\infty ...
2
votes
0answers
81 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
2
votes
1answer
1k views

Finding the taylor series of $f(z) = 1/(1+z^2)$.

I am working on the following exercise: Find the Taylor expansion of the function $f(z) = \frac{1}{1+z^2}$ about $z = 3i$. We had the Taylor Series Theorem in the lecture: Let $D \subset ...
3
votes
1answer
54 views

Counterexample: For real functions existence of all higher order derivatives doesn't imply analycity.

In the lecture we had an example for a function $f: \mathbb R \to \mathbb R$, which is not analytic. We defined, that a function is said to be analytic at some point $x_0$ if a Taylor series expansion ...
1
vote
1answer
44 views

f is a smooth function, and $M_n$ is the sup of $f^{(n)}$. Show if $\lim_{n \to \infty} \frac{M_n}{n!}R^n < \infty$, then f(x) is the taylor series.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function (i.e. assume that the n-th derivative $f^{(n)}$ is defined on all of $\mathbb{R}$). Let $R$ denote the radius of convergence of the Taylor ...
1
vote
0answers
29 views

Find the order of the following expression as x->0

Could someone help me find the order of the following expression without using the quotient rule? $\frac{1-\cos(x)}{1+\cos(x)}$ I expanded the denominator and the numerator but not sure how I get to ...
2
votes
1answer
36 views

Proving that 2 functions are equal/not equal

Prove the equality of $f_1$ and $f_2$ given the following conditions: Problem 1 $f_1(x)$ and $f_2(x)$ are functions of finitely summed sine and cosine functions (e.g. $3\cos2x+\sin5x$), any ...
1
vote
1answer
43 views

Maclaurin series of $\ln(2+x^2)$

Find the Maclaurin series of $\ln(2+x^2)$. I know that $\displaystyle\ln(1+x) = \sum_{n=1}^\infty\frac {(-1)^{n-1}} {n} x^n $ So is $\displaystyle\ln(1+x^2) = \sum_{n=1}^\infty \frac ...
8
votes
1answer
129 views

How can I compute this limit? [duplicate]

I have to compute $$ \lim_{n\to\infty} \exp(-n)\left(1+n+\frac{n^2}{2}+\ldots+\frac{n^n}{n!} \right)$$ I think the value is 1, but i don't know how to proof this. Do I have to estimate the remainder ...
0
votes
4answers
128 views

Find $a$ such that $\lim_{x\to 0} \frac{1-\cos(\sqrt{ax})}{x^2}=3$.

Find $a$ such that $$\lim_{x\to 0} \frac{1-\cos(\sqrt{ax})}{x^2}=3.$$ Can we solve it with l'Hospital's Rule or do we need to use Taylor series? I have tried using L'Hospital's Rule and i keep ...
5
votes
1answer
542 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) - ...
2
votes
0answers
78 views

Taylor series $\frac{\sin x}{x}$ convergence

I needed the Taylor series for $f(x) = \frac{\sin x}{x}$ in $a = 0$. I started with $ f(x) = \frac{1}{x} \cdot \sin(x) $, used the existing $sin$ Taylor series and multiplied by $\frac{1}{x}$: $$ ...
0
votes
1answer
44 views

Convergence of an analytic function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a smooth function. Let $R$ be the radius of convergence of the Taylor series centered at $a.$ For each $n \in \mathbb{N},$ let $M_n= \sup\{f^{n}(t) : t \in ...
2
votes
0answers
453 views

Compute the first five non-zero terms of the Taylor series about $a=4$ for $f(x)=\sqrt{x}$

This is my first Taylor Series problem and I want to make sure I completed it correctly. Here is the question: Compute the first five non-zero terms of the Taylor series about $a=4$ for ...
3
votes
3answers
261 views

Find an accurate value of $f(x)=\sqrt{4x^2+x}-2x$ for large values of x. Calculate $\lim_{x\to\infty}f(x)$

My works: $x^2$ can be very large if x is large, thus the function has lose-of-significance error and we need to reformulate it. $$ ...
1
vote
1answer
96 views

Use Taylor polynomials with remainder term to evaluate the following limits $\large\frac{e^x-x-1}{x^2}$

My work: Since $\large e^x=\sum\limits_{j=0}^\infty \frac{x^j}{j!}$, then $\large\frac{e^x-x-1}{x^2}=\sum\limits_{j=2}^\infty \frac{x^{j-2}}{j!}=\sum\limits_{d=0}^\infty \frac{x^{d}}{(d+2)!}$. (Let ...
2
votes
0answers
240 views

Taylor expansion: first derivative approximation with third order

About first derivative approximation with third order. Let $$f'(t)=\frac{(2t+h)\cdot{f(h)}-4t\cdot{f(0)}+(2t-h)\cdot{f(-h)}}{2h^2}+R.$$ Show that $$R=\frac{f'''(\xi)\cdot{(3t^2-h^2)}}{6}$$ and ...
1
vote
0answers
650 views

Taylor's Formula vs. Taylor's Inequality

In my calculus book, Essential Calculus, and in class we were using Taylor's formula to approximate the remainder in Taylor polynomials but I am having a bit of trouble understanding the intuition ...
2
votes
0answers
35 views

Optimization, descent direction, neccessary condition

I'm learning about nonlinear, unconstrained optimization. In my book it says that a descent direction $p_k$ must satisfy: $$p_k\nabla f(x_k)^T < 0$$ This seems to mean that $p_k$ must be obtuse to ...
1
vote
2answers
100 views

Series expansion of a function at infinity

I know it is possible to expanse an expandable fonction for a real, and for infinite by setting $x=\dfrac1y$ and then expanse for $0$. But my question is, how do we do if the evaluation of the new ...
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0answers
52 views

Taylor's Formula and 'z' values

I have attached Taylor's Formula, an exercise problem from a section on Taylor polynomials, and the solution to this exercise. I understand part a, expanding $f$ using Taylor polynomials is the ...
0
votes
1answer
47 views

Suppose $f \in C^{\infty}(\mathbb R)$ and $\lim_{n \to \infty} \frac{1}{n!} \int_0^a x^n f^{(n+1)}(a-x)dx=0.$ Show $f$ is analytic on $\mathbb{R}$.

Suppose that $f \in C^{\infty} (-\infty , \infty)$ and that $$\lim_{n \to \infty} \frac{1}{n!} \int_0^a x^n f^{(n+1)}(a-x)dx=0$$ for all $a\in \mathbb{R}$. Prove that $f$ is analytic on ...
1
vote
1answer
68 views

Second-order derivative wrt. vector

I have a scalar function $f(\mathbf{x})$, where its argument $\mathbf{x}$ is a vector. I am Taylor-expanding $f$, so I have to find $$ \mathbf{c}^2\frac{d^2}{d\mathbf{x^2}}f(\mathbf{x}) $$ where ...
2
votes
3answers
90 views

Proof that Polynomials Form a Basis

I'm not even sure this is a true statement, but can someone prove that the polynomials for a basis for continuous functions? This seems to be motivation for Taylor series, and several of the ...
4
votes
3answers
124 views

How to check the real analyticity of a function?

I recently learnt Taylor series in my class. I would like to know how is to possible to distinguish whether a function is real-analytic or not. First thing to check is if it is smooth. But how can I ...
1
vote
1answer
826 views

Taylor expansion of a vector field (notation question)

Is there an index-less notation (using gradiends, Jacobians, curls, hessians, anything) to describe a second-order term in the Taylor expansion of a vector field $\mathbf{f}(\mathbf{x}): \mathbb{R}^n ...
0
votes
2answers
60 views

Estimate Interval of Validity of $1-\frac{x^2}{2}$ for $\cos(x)$

I have been struggling with the following problem and was wondering if anyone could provide some insight or suggestions: Use $1-\frac{x^2}{2}$ as an approximation to $\cos(x)$, with an error not ...
1
vote
1answer
139 views

How to prove that a complex power series is differentiable

I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of ...
1
vote
1answer
79 views

Find “singular expansion” of a function

I have the function $(1-z)^{-z}$, analytic except on $\mathbb{R}_{\geq 1}$ Now in the text, it says the "singular expansion" at $z=1$ is $\displaystyle \frac{1}{1-z} + \log(1-z)+O((1-z)^{1/2})$ I'm ...
1
vote
1answer
26 views

Taylor polynomial aproximation - Interval of convergence

It is reaquired to find the Taylor polynomial of order $n$ of the cosine function around $x=0$. Then, it asks to find the biggest interval in which the sequence pf polynomial $p_n$ converges to $f(x) ...
2
votes
2answers
37 views

Does |Taylor Series of $f$ - $f$| Converge Monotonically to $0$?

Suppose that $T_n(x)$ be the sum of the first $n$ terms of the Taylor series of $f$ centered at $a$, and $\lim_{n\to \infty} T_n(b)=f(b)$. Is the difference $|T_n(b)-f(b)|$ decrease monotonically? ...
1
vote
1answer
103 views

Taylor theorem doubt(sin(x+h))

I was studying Taylor theorem when I came across this question in one of my math text books Obtain Taylor's series expansion of the function $\sin(\frac {\pi}{4}+h)$ in ascending powers of $h$. ...
2
votes
2answers
94 views

Taylor Series for $\log(x)$

Does anyone know a closed form expression for the Taylor series of the function $f(x) = \log(x)$ where $\log(x)$ denotes the natural logarithm function?
4
votes
2answers
211 views

Evaluating $\int_{0}^{\frac{\pi}{2}} \arctan( a \sin x) \ dx$ using the Taylor expansion of $\arctan (x)$

I was wondering if it's possible to show that for $a >0$, \begin{align}\int_{0}^{\pi/ 2} \arctan (a \sin x) dx &= 2 \sum_{k=0}^{\infty} \frac{\left(\frac{\,\sqrt{\vphantom{\Large A}\,1 + ...
1
vote
0answers
56 views

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 ...
3
votes
1answer
1k views

Find the Maclaurin series for $\cos(2x)$ using the series for $\sin(2x) $.

I know that $$\sin(2x)= 2x - \frac{8x^3}{3!} + \frac{32x^5}{5!} - \frac{128x^7}{7!} + \cdots $$ $$\sin(2x)= \sum_{n=0}^\infty (-1)^n {2^{2n+1}x^{2n+1} \over (2n+1)!}$$ But I don't see how I can use ...
1
vote
1answer
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 ...
1
vote
0answers
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 * ...
3
votes
2answers
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 ...