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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Is this part of a known sequence?

while trying to express as an infinite sum the function $t^x/\Gamma(x)$ I came across some coefficients of the form $a_0=1$ $a_1=-\psi^{(0)}(1)$ $a_2=[\psi^{(0)}(1)]^2-\psi^{(1)}(1)$ ...
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6 views

Find largest value of the constants that makes the equality true

$$sin(h^2) = a + O(h^b)$$ Find the largest value of a and b such that the equality holds. I tried to use the truncated Taylor series expansion of $sin(h^2)$, but the derivatives of the function are to ...
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8 views

Expansions and Approximations of Functions

When looking at series expansions of functions, the main cases I've worked with have been Taylor series and Fourier series. Whilst the latter can arguably be viewed as a subset of the former (e.g. ...
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2answers
18 views

Finding a root approach with a polynomial

So, i'm solving last's year's exams in Mathematical Analysis and i've found one interesting. It says: The equation $e^{-4x}=5x^2$ has one root close to (nearby) 0. By approaching $e^{-4x}$(close to ...
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1answer
34 views

Sum of $n6^{n}z^{n-1}$

I'm going mad calculating the sum $\sum_{n=0}^{\infty}n6^{n}z^{n-1}$. I proceeded in this way: $\frac{1}{z}\sum_{n=0}^{\infty}n(6z)^n$ and I'd like to figure out a geometric serie, but how can I take ...
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1answer
21 views

Function whose $n$-th derivative at $x=0$ is $n^3$ / evaluating the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$

I have troubles proving that the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$ represents the function $f(x)=e^x(x^3+3x^2+x)$. My idea was using the identity theorem for power series and the ...
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1answer
35 views

The derivatives of Riemann xi function

What are the first few values of derivatives of Riemann xi function at zero? Is there any general formula for calculating the nth derivative of the riemann zeta function at zero? What happens to the ...
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30 views

Find required degree of Maclaurin polynomial to estimate the cosine to two decimal places

I have a question where I am asked to find the amount of terms required in a Maclaurin polynomial to estimate $\cos(1)$ to be correct to two decimal places. So far what I have done is used Taylor's ...
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1answer
39 views

Infinite Sum Defined by $\int \frac{e^x}{x}dx$ vs. Exponential Function Taylor Series

Recently, when fiddling around with integration by parts, I noticed that it is possible to define infinite series that led to an integral. My calculus teacher noticed this, and told me to find $$ ...
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18 views

Taylor's expansion and remainder of $f(x)=0, -1\le x\le0$ and $f(x)=x^4, 0<x\le 1$

Let $f(x)=0, -1\le x\le0$ and $f(x)=x^4, 0<x\le 1$ If $$f(x)=\sum_{k=0}^n\frac{f^{(k)}(0)x^k}{k!}+\frac{f^{(n+1)}(\xi)x^k}{(n+1)!}$$ is the Taylor's formula for $f$ about $x=0$ with maximum ...
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2answers
81 views

Expansion $f(x)=1/(x-1)$

How to expand $f(x)=1/(x-1)$ into the form $1/x+1/x^2+1/x^3+...+1/x^n$ for x>1 I know f(x) can be rewritten as $f(x)=\frac{(1-1/x)^{-1}}{x}$. Next step is to expand $(1-1/x)^{-1}$ to ...
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1answer
36 views

Why is there no $-a\cdot t_0$ in the Taylor series?

Here is mine computation for uniform acceleration, i.e $v'(t) = \text{const}\ a$: $$v(t) = v(t_0) + \int_{t_0}^{t} v'(t)dt = v(t_0) + a\cdot \Delta t$$ where $$\Delta t = t- t_0$$ and coordinate is ...
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13 views

Disadvantages of Taylor series method

There is method called Taylor series method to solve non linear equations iteratively. I am interested to know ,what are the disadvantages of using this method to solve. General Idea any one please?
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2answers
35 views

Write the Maclaurin series for function $f\left(x\right)=\frac{1}{3x+1}\:$

We have function $f:\mathbb{R}\rightarrow \mathbb{R}$ with $$f\left(x\right)=\frac{1}{3x+1}\:$$ $$x\in \left(-\frac{1}{3},\infty \right)$$ Write the Maclaurin series for this function. Alright so ...
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1answer
41 views

Expansion of $\cos \sqrt x $?

If my $x$ belongs to non-negative real numbers then what is the expansion of $\cos \sqrt x$ . Is it sufficient to substitute $\sqrt x$ in place of $x$ in $\cos x$ expansion ? But I see ...
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10 views

Smoothing a function by subtracting terms in its Taylor series?

I am looking at some code for a Greens function that mentions the following % The GF is the smoothed by subtraction of first two odd Taylor series terms. So how does subtracting terms from a ...
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3answers
59 views

Find $\lim\limits_{x\to 0}\frac{\sqrt[]{\cos x}-\sqrt[3]{\cos x}}{\sin^2 x}$ using Maclaurin series

Applying the following expansions: $\sqrt[]{\cos x}=1-\frac{x^2}{4}+O(x^4)$ $\sqrt[3]{\cos x}=1-\frac{x^2}{6}+O(x^4)$ $\sin^2 x=x^2+O(x^4)$ gives the correct result on the limit $L=\frac{-1}{12}$. ...
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1answer
27 views

Showing that a function's Taylor series converges

Define $$\begin{align*} f(t) = \sum_{j=0}^\infty (-1)^j\binom{\frac{1}{2}}{j} t^j &= \sum_{j=0}^\infty (-1)^j \binom{2j}{j} \frac{(-1)^{j+1}} {2^{2j} (2j-1)} t^j \\ &= ...
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1answer
18 views

exponential of elementary matrix $\exp(tE_{a,b})$

$E_{a,b}$ is the elementary $n\times n$ matrix with $1$ in $(a,b)$-entry and $0$ elsewhere. Compute $\exp(tE_{a,b})$ for $a$ not equal to $b$. If $a=b$ then they would be on the diagonal, so ...
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2answers
35 views

What is the second derivative of $\sin(m \ln(x))$

I am trying to find a general Taylor Series for $y=\sin(m \ln(x))$ (where $m$ is a non-zero constant) however my answer conflicts with the answer in the worked example and I cant find the error in my ...
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1answer
11 views

Inequality in proof of Taylor polynomial for functions of 2 variables

I have this inequality that bothers me: \begin{equation} |g(h,x)|\le\bigl|o(\sqrt{h^2+k^2})\bigr|\cdot(|h|+|k|) \end{equation} where $g(h,k)=f(x+h, y+k)-f(x,y)-df(x,y)-\frac{1}{2}d^2f(x,y)$. So, ...
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1answer
18 views

Taylor Series in Fractional Calculus

I recently studied fractional calculus, namely the possibility to define fractional derivatives of some functions, like $$\frac{\text{d}^{1/2}}{\text{d}x^{1/2}}\ f(x) ~~~~~~~~~~~~~ ...
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26 views

Determining the minimal number of terms to use in a sum to approximate a number given a tolerance

In page 33-34 of Numerical Analysis by Burden & Faires an algorithm was given to compute the minimal value of $N$ for which $$|\ln{1.5}-P_N(1.5)|<10^{-5}\tag{1}$$,where ...
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2answers
38 views

Series expansion of $\frac{1}{\sqrt{x^3-1}}$ near $x \to 1^{+}$

How can I arrive at a series expansion for $$\frac{1}{\sqrt{x^3-1}}$$ at $x \to 1^{+}$? Experimentation with WolframAlpha shows that all expansions of things like $$\frac{1}{\sqrt{x^y - 1}}$$ have ...
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1answer
37 views

Finding the general Taylor polynomial formula for $f(x)=\log\left(\frac{1+x}{1-x}\right)$

I am trying to find the general form of the Taylor polynomial for $f(x)=\log\left(\frac{1+x}{1-x}\right)$. The $log$ is of base $e$ and I have rewritten the original formula as: $\log(1+x)-\log(1-x)$ ...
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45 views

A Taylor Expansion before Taylor

Taylor expansion was introduced in its currently well known form by Brook Taylor. Though the concept as this page says, has been formulated by James Gregory. Among his other works, Gregory established ...
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26 views

Finding a MacLaurin expansion of a function

I am being asked to Find the MacLaurin expansion of the following function: $f(x) = \frac{2x-8}{x^2-8x+12}$ I was not given a point about which to expand so I assume to use x=0. I know I can begin ...
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1answer
63 views

Taylor series for $x\over {\log(1-x)}$

Well, I understand expansions, but here I would like to have a general term for the coefficients in $$\frac x{\log(1-x)}=\sum_{n=0}^{\infty}a_nx^n$$ Direct calculations gives me ...
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1answer
18 views

Which approximation should I use?

I have a function $ k(x,y)$, and I want to approximate it for low values of x and y. $k(x,y) = \dfrac{a^3-ax^2-x^3+a^2x+ay^2-xy^2}{a^3-ax^2+x^3-a^2x+ay^2+xy^2}$ With $ a>>x, a>>y $ ...
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1answer
40 views

Finding general form of Taylor polynomial for function $f(x)=e^{x}\sin(x)$

I am trying to find the general form the Taylor polynomial of the function $f(x)=e^{x}\sin(x)$. I have calculated the derivatives up to $5$: $$\begin{align} f^{(1)}(x)&=e^{x}\cos(x) + e^{x} ...
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higher-order (3+) Taylor expansion of a likelihood function

I was wondering what is the effect if I replace the second derivative of the log-likelihood ("Likelihood" hereafter) function with its expectation in a higher-order Taylor expansion of the likelihood ...
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1answer
15 views

Multivariable taylor series expansion of $\exp(-(x^2+y^2))$

I stumbled upon the first-order taylor series expansion for multivariable functions described by $$T(x)=f(a)+(x-a)^T\nabla f(a)$$ and I wanted to expand $f(x,y)=\exp(-(x^2+y^2))$ around $(2,1)$ ...
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1answer
24 views

Series Expansion Relationship

In my textbook, The Physics of Vibrations and Waves - H.J. Pain pg. 28, it states the following : Writing $$ S(z) = 1 + z + z^2 + ... + z^{(n-1)} $$ and $$ z[S(z)] = z + z^2 + ... + z^n $$ we have ...
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52 views

Calculate $\ln(2)$ Without calculator using taylor series

I have to Calculate $\ln(2)$ Without calculator using taylor series. Help someone ? Thanks
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27 views

What does $\int \alpha(\alpha-1)f''(\alpha)d\alpha = f''(\phi)\int_0^1\alpha(\alpha-1)d\alpha$ has to do with taylor's theorem?

I'm reading a bad taken photo of the notes of my classmate, and there's an exercise that asks to calculate $\int_{0.2}^{0.4}\sin(x) dx$ by the fixed length trapezoid rule. Then there's this written ...
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49 views

Find the smallest value of $n$ for $P_n(x)$ to approximate $f(x)$ within $10^{-5}$ on $[-0.25,0.25]$

Let $f(x)=\ln(0.5+x)$ and let $P_n(x)$ be the $n$th Taylor polynomial for $f(x)$ about $x_0=0$. Find the smallest value of $n$ for $P_n(x)$ to approximate $f(x)$ within $10^{-5}$ on $[-0.25,0.25]$. ...
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37 views

Calculate the limit of $\lim _{x\to 0}\left(\frac{1-\frac{\left(1+x\right)^{\frac{1}{x}}}{e}}{x}\right)$

I have to calcuate this limit: $$\lim _{x\to 0}\left(\frac{1-\frac{\left(1+x\right)^{\frac{1}{x}}}{e}}{x}\right)$$ I have no idea how to start, maybe taylor series? Thanks.
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18 views

Expand function around point where function is zero

I have a function $f(x)$ that I want to expand around the point where $f(x)=0$, but finding this point is difficult. What is the general way to expand around the point $f(x)=0$ without actually ...
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29 views

$\ln (2)$ from Taylor series with error at most $10^{-5}$

How do I calculate $\ln (2)$ by Taylor series with error at most $10^{-5}$? I use Maclaurin: $$ \frac{(1+c)^{-(n+1)}\cdot {{(-1)}^n}}{n+1}<\frac{(-1)^{n}}{n+1}<\frac{1}{n+1}<10^{-5} $$ ...
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60 views

Understanding a taylor expansion

If $$s(y) = \begin{cases} 2\sin(y/2)/y & \text{if $y \neq 0$} \\ 1 & \text{otherwise} \end{cases}$$ why is the taylor expansion of $g(y) = \frac{1}{s(y)}$: $$ \frac{1}{s(y)} = 1 + ...
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3answers
67 views

Proving $\frac{1}{\sqrt{1-x}} \le e^x$ on $[0,1/2]$.

Is there a simple way to prove $$\frac{1}{\sqrt{1-x}} \le e^x$$ on $x \in [0,1/2]$? Some of my observations from plots, etc.: Equality is attained at $x=0$ and near $x=0.8$. The derivative is ...
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96 views

An Expression of $\ln(2)$

I saw online that the following infinite series has a value of $\ln(2)$: $\sum_{n=0}^\infty \left(\dfrac{1}{n+1}-\dfrac{1}{n+2} +\dfrac{1}{n+3}-\cdot\cdot\cdot\right)^2$ I started off by defining ...
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1answer
50 views

Bounding the remainder

I need to find the 3rd order Taylor polynomials and bound the remainder term at $(0,0)$. The function is $$f(x,y)=\cos(x)\sin(y)$$ Here is what I did: first, I found the taylor expansions of sin and ...
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1answer
25 views

Find the Taylor series and evaluate at $f^{39}(0)$

$$e^{-x^2}$$ I've had a hard time understanding power series since as long as I can remember. To my understanding, the question is asking me to write out the terms in the formula for Taylor series, ...
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1answer
100 views

Finding a Matrix of Rank 10 using Taylor Expansion

For $-1\leq t_i\le 1$ and $1\le i\le n$, we have an $n\times n$ matrix $A$ such that $$A_{ij}=\exp(t_it_j)$$ Now, how can we use the Taylor expansion of $e^x $ to find a $\text{rank }10$ matrix ...
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1answer
34 views

Cosine of matrix and matrix of cosines

Suppose I have cosine of the matrix $$ \tag 1 \cos\left( \begin{pmatrix} a & b \\ c & d\end{pmatrix}\right) $$ May I write it in a form $$ \tag 2 \begin{pmatrix} \cos(a) & \cos(b) \\ ...
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1answer
30 views

Let g(x) be the Mclaurin's expansion of sin(2x). If error is atmost $\frac{1250. 10^{-4}}{3} $ for x $\in$ $ [0,\frac{1}{2}]$

Let g(x) be the Mclaurin's expansion of sin(2x). If error is atmost $\frac{1250. 10^{-4}}{3} $ for x $\in$ $ [0,\frac{1}{2}]$ . Then minimum number of non zero terms in g is A.2 B.3 C.4 D.5 I ...
3
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1answer
116 views

Calculating ${(0.9)}^{\left(0.6\right)}$ with an approximation of ${10}^{\left(-4\right)}$

I'm having extreme difficulties understanding how to use Lagrange theorem to find an approximation. So far for my series I have: $$(1+(-x))^\frac{3}{5}= ...
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1answer
42 views

Calculate $\sin \frac \pi {10}$ with error bound of $10^{-4}$

I know there are similar posts about this question, but I have read them and it's still not clearly for me. I have to calculate $\sin \frac \pi {10}$ with error bound of $10^{-4}$. I know I have to ...
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0answers
22 views

Taylor series with mixed derivative

let $f: \mathbb{R}^2 \rightarrow \mathbb{R}: (x,y) \rightarrow f(x,y)$ a continuous, derivable function. Let $u := (x,y)$. Now, if you have $f(x+a,y+b) $with $a, b \in \mathbb{R}$ (and close enough to ...