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Questions tagged [taylor-expansion]

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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Taylor series expansion of $\frac{1}{\sqrt{1-\beta x(x+1)}}$

I am trying to find the taylor series expansion about $0$ (maclaurin series) of $$x \rightarrow \frac{1}{\sqrt{1-\beta x(x+1)}} \text{ with } \beta \in \mathbb{R}^{+*}$$ I've tried using the taylor ...
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1answer
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How to derive relation between relative and log deviations?

I ran into the following approximation: $$\frac{x_t - x_0}{x_0} \approx \log(x_t/x_0) + \frac{1}{2}\log(x_t/x_0)^2 $$ which is supposedly a second-order approximation around a fixed value $x_0$. How ...
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1answer
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ratiotest and radius of convergence of Taylor expansion of $x\sin(x)$

Today I helped a student who did not understand Taylor approximations. One of the exercises he had trouble with, was to determine the Taylor series of the function $$f: \mathbb{R} \to \mathbb{R}: x \...
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2answers
28 views

Taylor series converges to $\log\frac{1+x}{1-x}$ [on hold]

How can I demonstrate that $\sum^{\infty}_{n=1}\frac{2x^{2n-1}}{2n-1}$ converges to $\log\frac{1+x}{1-x}$ for every $|x|<1$?
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How to find the upper bound of the error in given Taylor polynomial?

We used this polynomial $P_2(x)=1-\frac{1}{2}*x^2$ to approximate the function $f(x)=\cos x$ in the interval $[\frac{-1}{2},\frac{1}{2}]$ How do I find the upper bound of the error?
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27 views

Prove that the function $f(x) = \sum_{k =1}^{\infty} \frac{\sin(kx)}{2^{k}}$ is infinitely differentiable

Prove that the function $f(x) = \sum_{k =1}^{\infty} \frac{\sin(kx)}{2^{k}}$ is infinitely differentiable This is a practice problem for an exam I have coming soon. I am trying to study, but I ...
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If the Taylor expansion of $f$ converges to $f$, prove that there are constants $C,R$ such that $f^{(k)}(x) \le C \cdot \frac{k!}{R^k}$

Q: Let $f$ be an infinitely differentiable function on an open interval $I$ centered at $a$. Assume that the Taylor expansion of $f$ about $a$ converges to $f$ at every point of $I$. Prove that there ...
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3answers
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Taylor series of $\ln\frac{1+x}{1-x}$ [duplicate]

Let $f(x)=\ln\frac{1+x}{1-x}$ for $x$ in $(-1,1)$. Calculate the Taylor series of $f$ at $x_0=0$ I determined some derivatives: $f'(x)=\frac{2}{1-x^2}$; $f''(x)=\frac{4x}{(1-x^2)^2}$; $f^{(3)}(x)...
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How to find the taylor series for $f(x) = \frac{1}{16-x^2}$ centered at 9 in summation notation

Having trouble finding the taylor series for the following function: $$f(x) = \frac{1}{16-x^2} \textrm{ centered at c=9}$$ I was trying to look at it in a way such that I could modify $\frac{1}{1-x}...
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1answer
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An even function, $f(z)$, analytic near 0 can be written as another analytic function, $h(z^2)$

If f is an even function, $f(z) = f(-z)$, and is analytic near 0, then there exists a function h, also analytic near 0, such that $f(z) = h(z^2)$ I suspect this statement is true because the ...
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2answers
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How can I show $\sum_{k=0}^{\infty} \frac{1}{1 + |2|^{k}}$ converges? [on hold]

How can I show $\sum_{k=0}^{\infty} \frac{1}{1 + |2|^{k}}$ converges? One way would be to just compute it, but I don't know how to deal with the absolute value. Additionally, I want to show that $\...
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Finite sum of $1/n^{\delta}$

How does one show $\sum_{n \leq x/M} \frac{1}{n^{\delta}} \approx \frac{(N/M)^{\delta}}{1-\delta}$, where $n \leq N$ and $\delta>0$. I assume this has something to do with Taylor series, but im ...
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3answers
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Finding Maclaurin’s series expansion function of $f(x) = a^x$ at $x=0$

First derivation is easy as it's $a^x * \log a$, but I have some troubles with finding the derivative of $ a^x * \log a$ Using the product rule I have $ (a^x *\log a)' = (a^x * \log^2 a) + a^x * \...
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1answer
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Existence of polynomial p such that $|f(x) − p(x^2)| < \epsilon$

Let $f$ be a real valued continuous function on $\left[−1, 1\right]$ such that $f(x) = f(−x)$ for all $x \in \left[−1, 1\right]$. Show that for every $\epsilon > 0$ there is a polynomial $p\...
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1answer
26 views

Taylor polynomial Approximation of a value using degree 3

How would I use a Taylor polynomial of degree 3 to approximate 33^(1/5). Can you please go in detail with your steps.
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4answers
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Proving that the power series for the cosine function is greater than zero, for $x$ in $[0, \pi/2)$.

I'm trying to prove the cosine power series $$\sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!} \;>\;0$$ for all $x \in [0, \pi/2)$. Here, $\pi$ is defined as the smallest positive real such that $...
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4answers
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Limit $\lim_{x\to\infty}\left(1-\frac{a^2}{x^2}\right)^{x^2}$

I found this example in a textbook, and I understand the author's reasoning and I also reached the same answer using L’Hôpital’s rule. However, I have two issues: Firstly: For any finite $a$, then ...
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3answers
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When Should I Use Taylor Series for Limits?

I get confused between when to apply L'Hospital Rule and Taylor Series. Is there any set of trigger points in the questions, that would be easier to solve with Taylor Series? For Example, If the ...
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1answer
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How do you expand a taylor series about a complex number?

Normally a Taylor series is constructed along real numbers. However, for practical purposes mathematics often heralds that commonly known continuous functions in the real plane are equivalent to their ...
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Question of a infinitely differentiable function [on hold]

I have tried using a few methods like Taylor series to solve the question but to no avail. Any help is appreciated.
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30 views

Find $\frac{\partial^{200}f}{\partial x^{200}}(0)$ if $\log(1+f(x)+f^4(x))= x^{50}$, $f(0)=0$ and f is differentiable infinitely many times at 0.

I understand that I need to somehow use Taylor series, but I have no idea what to do next. Can you give me a hint?
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4answers
270 views

Evaluating the limit using Taylor Series

We're asked to find the following limit by using Taylor expansions $$\lim_{x\to{}0}\frac{e^{3x}-\sin(x)-\cos(x)+\ln(1-2x)}{-1+\cos(5x)}$$ My Attempt: Expressing $e^{3x}$, $\sin(x)$, $\cos(x)$, $\ln(...
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0answers
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Why isn't the first term of the McLaurin series for $\cos(x)$ a pole? [closed]

As I understand it, the McLaurin series for $\cos(x)$ is $$\sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k)!}$$ This leaves me puzzled. Websites I've seen give the expansion of the sum as $$1-\frac{x^2}{...
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Expansion of $\sqrt{(a+3i)^2+x^2}$ with $x,a\in \mathbb R,\ x>>a, 3$

I know that if we only have $\sqrt{a^2+x^2}$ with $a,x\in \mathbb R,\ x>>a$, then we do the normal Taylor expansion around $\frac{a}{x}<<1$, but what if I want to expand something like $\...
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2answers
58 views

Calcul of $\lim_{x\to 1^{-}}\frac{\pi-\arccos(x)}{\sqrt{1-x^2}}$

I found that $$\lim_{x\to 1^{-}}\frac{\pi-\arccos(x)}{\sqrt{1-x^2}}=+\infty$$ My question: Can we use Taylor series method to find this limit?
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Taylor expansion of likelihood function

$\require{\cancel}$ ...For large samples, as a consequence of the central limit theorem, the likelihood function approaches a gaussian, whose expected value is equal to the maximum likelihood ...
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109 views

Taylor series with a base point different from $0$

What's the need for $f(x) = \sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{k!}(x-a)^{k}$ if we already have the formula at $0$? Isn't the $(x-a)$ just making the $a$ as the new origin? When is this formula more ...
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1answer
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The conditions under which the Taylor polynomial $P_n(x)$ will converge to $f(x)$

Background It is well-known that for the following function: $$f(x) = \begin{cases} e^{\frac{-1}{x^2}} & \text{if} & x \neq 0 \\ 0 & \text{if} & x = 0 \end{cases} $$ the Taylor ...
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1answer
24 views

Functional Gradient Descent and Functional Taylor Expansion

The questions are based on the below screenshots. Can somebody explain how the functional Taylor expansion is related to a "standard" function Taylor expansion? In particular, I am concerned with ...
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1answer
18 views

Higher Order Multivariable Taylor Expansions

The quadratic multivariable Taylor approximation of a function $f(x, y)$ around a point $(a, b)$ is given by $f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) + \frac{1}{2}f_{xx}(a, b)(x - a)^2 + f_{xy}(...
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2answers
44 views

Laurent Series expansion about the point $z_0 = i$ of $\frac{z}{z^2+1}$

I am trying to construct the Laurent series expansion of $f(z) = \frac{z}{z^2+1}$ about $z_0 = i$ in the region $\{z \in \mathbb{C}: 0 < |z - i| < 2\}$ but I am stuck. We can re-write $f(z) = \...
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2answers
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Lagrange notation: $f^{(0)}(x)$?

Using Lagrange notation, is $f^{(0)}(x)=f(x)$? Is this standard notation, or would one have to define $f^{(0)}(x)=f(x)$ first, before using it? Aside: the context of the question is whether to ...
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1answer
43 views

Combinatorial Proof that the Logarithm of a Product is the Sum of the Logarithms

I've been strongly drawn recently to the matter of the fundamental definition of the exponential function, & how it connects with its properties such as the exponential of a sum being the product ...
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1answer
24 views

Taylor expansion on Images which are warped

I'm currently reading following paper: https://www.ri.cmu.edu/pub_files/pub3/baker_simon_2001_2/baker_simon_2001_2.pdf Here I struggle to follow the Taylor expansion from equation 2 to equation 3. I ...
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Why does $\sum_{n\geq0}(1-x)^n=\frac1x$ have such a poor radius of convergence?

I am confused as to why $$\sum_{n\geq0}(1-x)^n=\frac1x$$ only works for $x\in (0,2)$. I get that it has a singularity at $x=0$, so that can't work, but there are no singularities for the rest of the ...
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1answer
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Prove $y^tH_f(a)y \leq 0$ with Taylors Theorem

Let the function $f \in C^2(\mathbb{R}^n;\mathbb{R})$ have a local maximum in the point $a \in \mathbb{R^n}$. How can one prove the following with Taylor's theorem: The following applies: $y^tH_f(a)...
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1answer
18 views

Simplifying $\sum_{n=1}^{x}ne^{-a}\frac{a^{x-n}}{(x-n)!}$, where $x$ is an integer and $a<1$

I would like to simplify the following expression, $$\sum_{n=1}^{x}ne^{-a}\frac{a^{x-n}}{(x-n)!}$$ where $x$ is an integer and $a<1$. Is it possible to lose the sum? An approximation for the ...
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1answer
31 views

Using the Cauchy-Hadamard Theorem to find a radius of convergence

Cauchy-Hadamard Theorem Consider the formal power series $$f(z) = \sum_{n = 0}^{\infty} c_{n}(z - a)^{n} $$ for $a, c_{n} \in \mathbb{C}$. Then the radius of convergence of $f$ at the ...
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Find the limit of the expression $\lim_{x\to 0}\left(\frac{\sin x}{\arcsin x}\right)^{1/\ln(1+x^2)}$

Limit: $\lim_{x\to 0}\left(\dfrac{\sin x}{\arcsin x}\right)^{1/\ln(1+x^2)}$ I have tried to do this: it is equal to $e^{\lim\frac{\log{\frac{\sin x}{\arcsin x}}}{\log(1+x^2)}}$, but I can't calculate ...
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1answer
32 views

Prove $f^{(n)}$ exists on $\mathbb{R}$ and $f^{(n)}(0)=0$ for all $n$, where $f(x)=e^{\frac{-1}{x^2}}$

$f(0)=0$ and $f(x)=e^{\frac{-1}{x^2}}$ for $x\neq 0$. I need to prove $f^{(n)}$ exists on $\mathbb{R}$, $f^{(n)}(0)=0$ for all $n$, and every Taylor Polynomial about $0$ is $0$. After setting up the ...
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3answers
62 views

Simplifying $\sum_{n=0}^{\infty}ne^{-a}\frac{a^{x+n}}{(x+n)!}$, where $x$ is an integer and $a<1$

I would like to simplify the following expression, $$\sum_{n=0}^{\infty}ne^{-a}\frac{a^{x+n}}{(x+n)!}$$ where $x$ is an integer and $a<1$. Is it possible to lose the sum? An approximation for ...
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1answer
25 views

Find 3rd order Taylor polynomial

Find the 3rd order Taylor polynomial of $f(x,y)=\log(1+x-y)$ at $a=(0,0)$ So I'll just list out the partials I computed: $f_x=\frac{1}{x-y+1}$, $f_y=\frac{-1}{x-y+1}$, $f_{xx}=\frac{-1}{(x-y+1)^2}$. ...
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1answer
48 views

Why does infinite sum series of cos k diverge? [closed]

So I just finished a coursework for maths and concluded that the infinite sum of $\cos k$ from $k=0$ is a convergent one using the Maclaurin series expansion and doing the ratio test for which i got $...
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1answer
22 views

Radius of convergence for $\ln(a+x)$

Since the radius of convergence $R$ for the Taylor series of $\ln(z)$ around $1$ is $1$, i.e. $$ R\left\{\ln\left(1+z\right) = \sum \frac{(-1)^{k-1}}{k}x^k \right\} = 1 ,$$ does this mean that for $...
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2answers
69 views

Calculate $\sum\limits_{n=0}^{\infty} \frac{x^{3n}}{(3n)!}$ [duplicate]

$$\sum\limits_{n=0}^{\infty} \frac{x^{3n}}{(3n)!}$$ should be calculated using complex numbers I think, the Wolfram answer is : $ \frac{1}{3} (e^x + 2 e^{-x/2} \cos(\frac{\sqrt{3}x}{2})) $ How to ...
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1answer
20 views

Calucate the coefficient of taylor series

Suppose $f$ is a real valued function with $f(0) = 0$ and $f'(x) = 1/\sqrt{1+x^2}$. I want to calculate the coefficient of the $x^7$ term around $0$. My approch: Although I know the answer if given $$...
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1answer
13 views

Determination of the order of a pole

In the function $$f(z) =\frac{sin(\frac{\pi}{2}(z+1))}{(z^2+2z+4)(z+1)^3}$$ the order of the pole in $z=-1$ is correctly two? Or maybe it is an eliminable singularity? I have a problem because often ...
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0answers
9 views

Approximation around steady state:

All equations here are direct copies from a textbook. In steady state, we have $$i = \rho + \pi + \sigma y.$$ The goal is then to do a first-order Taylor approximation around this steady state of ...
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1answer
46 views

Bounded function with bounded second derivative imply bounded first derivative

Let $f$ be a $C^2$ function from $(t_1,t_2)$ to $R^n$ such that $\Vert f(t)\Vert\leq A$ and $\Vert f''(t)\Vert \leq B$ for all $t\in (t_1,t_2)$, where $A$ and $B$ are nonnegative reals. Let $t_0\in (...
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1answer
35 views

Finding the Maclaurin series of $e^{\sin x}$ by comparing coefficients

I believe I have found a nice way to find the Maclaurin series of $e^{\sin x}$. Please check if there are any mistakes with my working. Is this method well known?