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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To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
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2answers
37 views

Compute Taylor Series

For the question above I have done the first few Taylor series calculations; they are below. Now I am finding it difficult to transform these terms into a series. Every equation I come up with is ...
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1answer
85 views

Can we use taylor series to solve difficult equations (example with cos(x)=x)?

Well I saw that the curve of the taylor function series of $\cos(x)$ at $x=0$ marry (it's a french expression to say that is very very near) the curve of $\cos(x)$ between $x=0$ and $x=pi/2$ So if I ...
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0answers
26 views

please help me completing this proof (Lagrange remainder for Taylor formula)

I'm trying to prove that the remainder of a $n$-th grade Taylor formula is $$R_n=\frac{f^{(n+1)}(\mu) (x-x_0)^{n+1}}{ (n+1)!}$$ where $\mu$ is a value between $x$ and the centre $x_0$. For $n=1$ it ...
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38 views

Compute the first four terms of the Taylor Series

"By multiplying the appropriate Taylor series about $c=0$, compute the first four terms of the Taylor series about $c=0$ for $f(x)=e^{-x}\cos x$." Seems straightforward enough but when I break up ...
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0answers
95 views

Find the function whose Taylor series is $\log(x)+\log(x+1)+\log(x+2)+\ldots$

How do I find a function $f$ whose Taylor series is $$\log(x)+\log(x+1)+\log(x+2)+\ldots$$ for some point $x=a$? It would seem that $$\left.\frac{\partial^n}{\partial x^n}f(x) \right|_{x=a} = ...
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5answers
98 views

Taylor expansion of $\frac{1}{1+x^{2}}$ at $0.$

I am trying to find the Taylor expansion for the function $$f(x) = \frac{1}{1+x^{2}}$$ at $a=0.$ I have looked up the Taylor expansion and concluded that it would be sufficient to show that ...
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1answer
47 views

Understanding the proof of Taylor's theorem

I'm trying to understand the proof of Taylor's theorem from here: I already made a question about the remainder part of the theorem and got an answer for it here: Remainder term in Taylor's ...
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1answer
55 views

Remainder term in Taylor's theorem

I'm trying to understand the remainder in Taylor's theorem from this source: https://proofwiki.org/wiki/Taylor%27s_Theorem/One_Variable/Integral_Version I don't understand the very last parts of ...
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1answer
29 views

Aftermath of Cauchy's mean value theorem

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
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1answer
19 views

Finding the convergence radius of a complex laurent series

Find the maximal ring where the following series converges: $$\sum_{n=1}^\infty\frac{3^n+2^n}{(z-5)^n}+\sum_{n=0}^{\infty}\frac{n^2}{20^n}(z-5)^{2n}$$ I think that taking the minimum between the ...
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1answer
14 views

Question about Peano form of the remainder

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
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3answers
51 views

Taylor Series for $(1-x)^p$

Can anybody help me with the Taylor series for $(1-x)^p$? I have no idea how to do it. I know that: $(1-x)^{-1}=1+x+x^2+x^3+...$ Any help would be much appreciated.
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0answers
12 views

Taylor's expansion of the singular part of an analytic function

Assume $f$ is analytic on the annulus $R_1<|z-a|<R_2$. Assume $R_1<r<|z-a|$. Define $f_2$ by $$f_2(z)=\frac1{2\pi i}\int_{|x-a|=r}\frac{f(x)dx}{x-z}$$ $f_2$ is analytic on $|z-a|>r$. ...
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0answers
12 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
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1answer
21 views

Algebraic Proof of Sum of Exponential Powers is Product of Exponentials

Can somebody provide a proof of the summation of powers law for the product of two exponentials, using only algebra and the Taylor series, no derivatives or calculus tricks?
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1answer
47 views

Convergence of Taylor series of $\sqrt{1-x}$

Concerning $$\sqrt{1-x} = \sum_{k=0}^{\infty} \left[\prod_{j=1}^k \left(\frac{j-1-\frac{1}{2}}{j}\right)\right]x^k$$ the Taylor series about $x=0$. For $|x|< 1$ this series converges uniformly. ...
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1answer
23 views

Taylor expansion of the electrostatic potential $1/\|\cdot \|$

I have stumbled over this problem several times in electrodynamics, and I just don't get the hang of it. The task is to do a Taylor expansion of $\,f(\vec{x},\,\vec{a}) = ...
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1answer
34 views

How do you represent f(x+h) and f(x−h) as a Taylor series using the taylor series formula?

I know the answers are below, however i am not quite sure what to substitute as the "a" in the Taylor series formula. $f(x+h)=f(x)+f′(x)⋅h+\frac 12f′′(x)\cdot h^2+\cdots+\frac 1{n!}f^{(n)}(x) \cdot ...
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24 views

Two case about convergent series

Could you help me to prove analytically that ? I started to study Taylor Series and I'm lost. Thanks in advance.
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1answer
14 views

Re-expressing the Schrodinger Equation as a first order expansion.

I am reading an online text on quantum computing and the author expands and re-expresses the Schrodinger equation. I am not really sure as to the intermediate steps he used or what happened to the ...
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1answer
39 views

Conditions must satisfy $f: (a, b) \to \mathbb{R}$ so that its Taylor series converge to f itself. [duplicate]

I have a doubt. What conditions must satisfy $f: (a, b) \to \mathbb{R}$ so that its Taylor series converge to f itself.
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5 views

Taylor expand an multivariate function about singularity

For example, is it possible to Taylor expand the following function: $$ f(x,y) = c \sqrt{ \frac{x^2}{4 a^2} + \frac{y^2}{a^2} } $$ about $(x,y) = (0,0)$?
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2answers
179 views

Solution to curious infinite series

How exactly does one find a closed form to: $$ \sum_{i=0}^{\infty}\left[\frac{1}{i!}\left(\frac{e^2 -1}{2} \right)^i \prod_{j=0}^{i}(x-2j) \right]$$ When expanded it takes on the form $$1 + ...
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0answers
44 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $∑_{n≥0}q^{n(n+1)/2}(1+q)(1+q^2)...(1+q^n)u^n$ where both q and n are variables and $n \in N∪0$?
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5answers
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Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
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1answer
34 views

Derivatives as Linear Approximations

I have always thought of the fact that a derivative is a linear approximation as being nothing more than that- an approximation. But is there an epsilon-delta meaning behind that? Is there a stronger ...
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2answers
76 views

Taylor series of the function $f(x) = (1+x) ^{\frac{1}{x}}$

Good night!! I got this problem and I'd like to find all the mistakes in my statement. This is my prblem. Find the binomial coefficients of $f(x) = (1+x)^{\alpha}$, with $\alpha \in \mathbb{R}$, and ...
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1answer
66 views

How do I construct such a numerical method for solving ODE?

I am asked to expand $x(t+h)$ and $x(t+2h)$ around $t$ up to the rest term of the third order, find $A, B, C \in \mathbb R$ such that $$x'(t)=\frac{Ax(t)+Bx(t+h)+Cx(t+2h)}{h} + O(h^2)$$ and based on ...
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1answer
23 views

How to approximate the bounding region of a 2d differentiable mapping locally?

I have got a differentiable mapping $f:\Bbb R^2 \to \Bbb R^2$, Is the image of $f$ of a very small convex subset (e.g., a unit square) around any point, a bounded region? If it is bounded, can I ...
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2answers
48 views

Substituting for Taylor series

So my question is simple: Why is substitution valid? I mean it seems counter-intuitive to me mainly because of the chain rule. For example: The Taylor series of $e^{x^2}$ is simply done by ...
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0answers
23 views

Taylor polynomial, Peano form of the remainder of f(x) and its asymptote

Could someone help me complete this or check if my reasoning so far is correct? I'm stuck at finding the oblique asymptote: Write the Taylor polynomial and the Peano form of the remainder of $f(x) = ...
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Basic doubt about derivative

given the following equation: $f(x) = (x-1)\arctan x$, I am to calculate its Taylor polynomial of 2nd degree with remainder of Peano. I began by calculating the derivative, using the product rule: ...
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2answers
38 views

Does $\sum\limits_{n=1}^\infty \left[n\left(f\left(\frac{1}{n}\right)-f\left(-\frac{1}{n}\right)\right)-2f'(0)\right]$ converge?

Let $f\in C^3([-1,1])$ Is the series $\sum\limits_{n=1}^\infty \left[n\left(f\left(\frac{1}{n}\right)-f\left(-\frac{1}{n}\right)\right)-2f'(0)\right]$ convergent? I'm trying to use Taylor's ...
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2answers
33 views

How to find the exponent $z$ of $(-1)^z$ for a patterned series of signed ones?

A question in Larson "Calculus" asks for the Taylor series centered at $\frac{\pi}{4}$ of $\cos(x)$. This expands to: $a_n = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) - ...
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0answers
26 views

Entire periodic $f(z)$ with more than 50 % of the derivatives $0$?

Im looking for a real-entire function $f(z)$ such that for any complex $z$ : $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number. $2)$ $f(z)= 0 + a_1 z + a_2 z^2 + a_3 z^3 + ...$ where more than ...
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20 views

Basis for a general function of tr$X^k$

I have a function of a $4\times 4$-matrix variable $X$ which is a general function of $\left<X^k\right>$ where $\left<\cdot\right>$ denotes the trace. My question is this: is there a ...
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3answers
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Prove that $\lim_{x\to\infty} f'(x) = 0$ [duplicate]

Let $f(x)$ be twice differentiable on $(0,\infty)$ and let $\lim_{x\to \infty} f(x) = L<\infty$ and $|f''(x)| \le M$ for some $M>0$. Prove that $\lim_{x \to \infty} f'(x) = 0$. I've tried to ...
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0answers
24 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
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1answer
29 views

Taylor's series and the function argument dimension

I've stumbled over an interesting question. In $\cos(x)$, $x$ is measured in, say, radians. When I expand cosine in Taylor's series, I have the terms with $x^3$, $x^5$ etc. so I am summing up ...
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26 views

Looking for a family of entire functions.

Let $i$ be a positive integer and let $f_i(x)$ be the $i$ th real transcendental entire function that has the Taylor expansion $f_i (x) = a_{i_0} + a_{i_1} x + a_{i_2} x^2 + ...$ with all $a_{i_n}$ ...
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1answer
29 views

Geometric interpretation of a Taylor series like identity

Johann Bernoulli published (something like) the following expression in his journal Acta Eruditorum. $\int_0^x f(t) dt = xf(x)-\frac{x^2}{2!}f'(x)+\frac{x^3}{3!}f''(x)-\frac{x^4}{4!}f'''(x)+...$ Is ...
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24 views

$X\sim\mathcal N(0,1)$, Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd?

If $X\sim\mathcal N(0,1)$ Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd ? ($\Phi_X^{(j)}(0):j^{th}$ derivative of the characteristic function of the r.v. $X$) We computed ...
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1answer
24 views

Approximating error using Taylors theorem

I have used a Maclaurin series for the function $f(x) = \cos(2x)$ and have successfully produced: $\dfrac{2^n cos(\frac{n\pi}2)x^n}{n!}$ Now I want to estimate the error in approximating $\cos(2x)$ ...
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2answers
24 views

How to show MacLaurin series for $\frac{1}{1-x}$ converges using remainder term

$\dfrac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots$ this is valid for $x$ between $-1$ and $1$ not including the endpoints. How can one show that the Lagrange remainder goes to zero as $n$ goes to ...
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1answer
14 views

First Order Approximation Taylor Series

I have the taylor series $f(z)=f(x_0)+(x-x_0)f'(z)+1/2(x-x_0)^2f''(z) ...$ and I am told that "As a first order approximation," $x-x_0$ ~ $\frac{f(x)-f(x_0)}{f'(x_0)}$ assuming $f'(x_0) \neq 0$ I ...
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2answers
45 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
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1answer
49 views

Let $f:[-1,1] \to \mathbb{R}$ be differentiable 3 times, prove $\exists M>0 \ , \ s.t \ f(x) \le Mx^2$

Let $f:[-1,1] \to \mathbb{R}$ be differentiable 3 times, let $f(0)=0$ and $f(x) \ge 0 \ \forall x \in [-1,1]$. Prove: $\exists M>0 \ , \ s.t \ f(x) \le Mx^2$. I separated the proofs to ...
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1answer
42 views

Finding Laurent series where given annulus is not in a singularity

I'm given a problem where I need to calculate the Laurent series of $f(z)$ inside the given annulus $$ f(z) = {1\over z^3(z-1)}; \quad 1 < |z| < 2 $$ From online resources(videos, notes) I ...
2
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1answer
23 views

Using Taylor's series in imporper integrals

Is it possible to simplify an improper integral using Taylor's series? How can I prove this procedure is correct? For example, take $$f(\alpha)=\int_0^{\infty} ...