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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1answer
48 views

Solution of $d^2u/dx^2 + u/A = 0 \ (\text{or } \ C),$ with conditions

Does the following ODE: $$d^2u/dx^2 + u/A = 0 \quad (\text{or } \ C),$$ have a solution with the conditions: $$ \left.\frac{d^2u}{dx^2}\right|_{x=0} = 0, $$ $$u(x=0) = B$$ and $$ ...
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1answer
72 views

Evaluate a limit using Taylor series

Let $$\lim\limits_{x\to 0}\frac{({\ln(1+x) -x +\frac{x^2}{2})^4}}{(\cos(x)-1+\frac{x^2}{2})^3}$$ Now, I know that I should utilize Taylor polynomial. $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ...
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2answers
41 views

Multplication of series

My textbook is taking about the Cauchy product and I don't quite understand it and it says that when multiplying series, the sum of the third one is equal to the product of the sums of first two ...
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0answers
26 views

Relationship between Lagrange interpolation and Taylor expansion

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h_{-1}$ and $x_{1} = x_0 + h_1$ with $h_1, h_{-1}$ > 0. Given a smooth function f, and an approximation to $f'(x_0)$ given by the ...
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1answer
37 views

Show that the approximation to $f$'($x_0$) has discretization error $O$($h^2$)

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h$ and $x_{1} = x_0 + h$ with $h$ > 0. Given a smooth function f, show that the approximation to $f'(x_0)$ given by the centered ...
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1answer
73 views

Taylor series and Lagrange's remainder f(x)=$e^x$

In my textbook the Lagrange's remainder which is associated with the Taylor's formula is defined as: $R_{n}(x)= \frac{(x-a)^n}{n!} f^{(n)}(a + \vartheta (x-a))$, for some $\vartheta$ $\in$ <0 ,1> ...
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0answers
75 views

Relationship between $\sum\limits_{n=0}^\infty \frac{a_n x^n}{n!}$ and $\sum\limits_{n=0}^\infty \frac{a_n^2 x^n}{n!}$

For an analytic function with the property $f^{(n)}(0)=a_n$, we have $f(x)=\sum\limits_{n=0}^\infty \frac{a_n x^n}{n!}$. This can be extended to $f^{(n)}(x)=\sum\limits_{n=0}^\infty \frac{a_{n+1} ...
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1answer
86 views

Approximation for lower incomplete gamma function

Does any one know approximations for the lower incomplete gamma function $\gamma(a,bx)$. The problem is this: I want to find the quantile function for the CDF of the gamma distribution. The CDF of the ...
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1answer
71 views

Calculating Laurent Series of Complex Function

How does one alternate the Bernoulli number series expansion $$\frac x{e^x - 1}=\sum_{n=0}^{\infty}\frac{B_nx^n}{n!}$$ To calculate the Laurent Series centered at 0 in the annulus of convergence of ...
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0answers
35 views

Laurent Series expansion with |z-1|

Here's the problem: Expand $\dfrac{e^z}{z-1}$ in a Laurent series convergent in $0 < |z-1| < \infty$. And here is my attempt, I just wanna know if it's right cause it seems kinda simple. ...
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0answers
16 views

Proving Taylor Series Estimation of Integrals

Given that $f, g, h$ are all continuos on $x \in [a,b]$, $g(x)$ approximates $f(x)$ with an error of at most $h(x)$ Meaning $|f(x) - g(x)| \le h(x)$ Have to prove $|\int_a^bf(x)\,dx - ...
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1answer
32 views

How many terms of the Taylor expansion should I develop?

How do I know how many terms of the power series should I develop to evaluate a limit? Example: Given this limit: $L:=\displaystyle\lim_{x\to0}\left(\frac{\ln(1+x)}x-e^{-x/2}\right)\frac1{\cosh x - ...
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2answers
41 views

Two-dimensional Taylor linearisation

I have to perform a first order taylor expansion of a function $f(\vec{x}) = f(x+u,y+1)$ at the point $\vec{a} =(x,y)$. My solution reads $$ f(\vec{x}) \approx f(x,y) + \left( \begin{matrix} ...
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1answer
60 views

Showing Taylor Series for $f(x) = e^{-x^2}$ converges to $f$

Show Taylor Series for $f(x) = e^{-x^2}$ converges to $f$ I am stuck because when taking the (n+1) th derivative of f, I do not see a general pattern. Meaning I am having difficulty in bounding ...
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1answer
19 views

Showing uniform continuity of function giving radius of convergence

Let $f$ be an analytic function on an open disk $D$ and let $R(z)$ denote the radius of convergence of the power series of $f$ about a point $z$. Is there an easy way to show that $|R(z_1) - R(z_2)| ...
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0answers
34 views

PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial ...
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1answer
86 views

Taylor Series Theorem

So I see the argument presented in taylor series, that $$\sum c_n (x-a)^n = \sum \frac{f^{(n)}(a)}{n!} (x-a)^n$$ or $c_n = f^{(n)}(a)/n!$ if $x=a$ the question is, since the above only holds when ...
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2answers
55 views

Finding $\lim_{x\to0} \frac{\sin(x^2)}{\sin^2(x)}$ with Taylor series

Evaluate $$\lim_{x\to0} \frac{\sin(x^2)}{\sin^2(x)}.$$ Using L'Hospital twice, I found this limit to be $1$. However, since the Taylor series expansions of $\sin(x^2)$ and $\sin^2(x)$ tell us that ...
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1answer
50 views

Function whose power series coefficients contain logarithms

Is there a function that can be expressed as a power series $$f(x)=\sum_{n=0}^\infty a_n x^n$$ whose coefficients $a_n$ are expressions containing $\log n$ or something similar?
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1answer
32 views

Estimating the remainder of Taylor series written in Lagrange form

Given the function $$f(x) = \ln\left(\frac{1+x}{1-x}\right)$$ Show that the error $f(1/3) - T_n(1/3)$ is at most $55/7776$ My attempt Remainder Term = $[f^{(5)}(x) = 24/(1+x)^5 + ...
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3answers
631 views

Why does this infinite series equal one?

Why does $$\sum_{k=1}^\infty \binom{2k}{k} \frac{1}{4^k(k+1)}=1$$ Is there an intuitive method by which to derive this equality?
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2answers
89 views

taylor series of ln(1+x)?

Compute the taylor series of $ln(1+x)$ I've first computed derivatives (upto the 4th) of ln(1+x) $f^{'}(x)$ = $\frac{1}{1+x}$ $f^{''}(x) = \frac{-1}{(1+x)^2}$ $f^{'''}(x) = \frac{2}{(1+x)^3}$ ...
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0answers
30 views

Tangent Line Approximation Using Taylor Series

I am new to Taylor series as a whole and was wondering if someone with a bit more background could validate my thought process in answering the following question. Question: Does the tangent line to ...
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1answer
89 views

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
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2answers
76 views

Differentiating the Taylor expansion of $e^x$

It is well known that a) $\frac{d}{dx}\exp x = \exp x$ and b) $\exp x = \sum\limits_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{3} + ...$. Therefore, it should be possible to ...
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0answers
51 views

Taylor series expansions of f(ax, y + dy)

I'm required to do a Taylor series expansion of $f(ax, y+ \delta y)$ where $a$ is a constant and $\delta y$ is an increment of $y$. How would it be done? This is probably a special case of the ...
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2answers
36 views

upper bound for the series $S (x) = \sum_{k=1}^{\infty} \frac{(x_n-n)^k}{k!}$ from $|x_n -(n+1)|\leq x$.

I've been trying to find a tight upper bound for the series $$S (x) = \sum_{k=1}^{\infty} \frac{(x_n-n)^k}{k!}$$ in terms of finite value $x\in \mathbb R$, where: 1- $\{x_n\}$ is a sequence of a ...
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2answers
49 views

How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
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2answers
176 views

Multiple differentiability from Taylor expansion

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a real function, and let $0\leq n\leq+\infty$. We make the following assumption: For every $a \in\mathbb{R}$ and for $k=n$ (resp., in the case $n=+\infty$: ...
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0answers
64 views

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
92 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
34 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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0answers
70 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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5answers
112 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
62 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
68 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
37 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
25 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
19 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
54 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
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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
23 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
80 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
63 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
26 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
38 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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0answers
25 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
19 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 ...