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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55 views

Find $\lim_{x\to\infty} (x \ln(16x +14) - (x \ln(16x +7))$ using Maclaurin series.

I am trying to find the limit of $\lim_{x\to\infty} (x \ln(16x +14) - (x \ln(16x +7))$. I know I have to use Maclaurin series, but something went wrong.
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1answer
16 views

Make $f(x)=\sin x-\frac{x+ax^3}{1+bx^2}$ be the infinitesimal of the highest order

Here is the question: Find $a$ and $b$, letting $$f(x)=\sin x-\frac{x+ax^3}{1+bx^2}$$ be the infinitesimal of the highest order when $x \to 0$, and find that order. According to the key, ...
3
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0answers
68 views

How to find the Maclaurin series for the integral of $e^{x^2}$?

I am trying to find the Maclaurin series for the integral of $e^{x^2}$? What I done so far is that the Maclaurin series for $e^{x^2}$ is $$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$$ So would ...
3
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1answer
57 views

Would the order of Taylor Polynomial change after substitution?

I found the order of Taylor Polynomial is kind of confusing. For example, we know: $$T_4e^x = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \frac {x^4} {4!}$$ After substitute $x$ as $t^2$, we ...
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0answers
34 views

Maclaurin series for exponential function

I have a question about maclaurin series for exponential functions for : Normally, $\frac{f'(0)}{1!}x =3x^2\times xe^{x^3}=3x^3e^{x^3}$ not $x^3$ because
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0answers
55 views

Does the radius of convergence always exist?

Does the radius of convergence always exist? For $\sum_{n=0}^{\infty} a_n (x-c)^n,$ my textbook states that the radius of convergence is $R=1/\limsup \sqrt[n]{|a_n|}$. However, what if this limit ...
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1answer
24 views

Sine squared when obtaining a Maclaurin and taking it's limit as $x\to 0$

When determining the Maclaurin series for $\sin^2(x)$ we use the trigonometric identity $\frac{1-\cos(2x)}{2}$. But when taking the limit of e.g. the Maclaurin series for $\cos(\sin(x))$ as $x$ ...
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2answers
46 views

Show that the radius of convergence of $e^x$ is infinite

I am a bit confused as to whether I am doing this question correctly. Firstly, we have defined the radius of convergence of a power series centered at a $$\sum_{n=0}^{\infty} a_n(x-a)^n$$ to be the ...
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6answers
6k views

Intuition explanation of taylor expansion?

Could you provide a geometric explanation of taylor expansion?
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1answer
23 views

Evaluating Maclaurin Series

I would like to know how they got the highlighted part in the image below; What I have done so far is, finding the Maclaurin series for $e^x$ then substitute $2x$ for $x$ and find the Maclaurin series ...
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0answers
22 views

Compute the limit of the error term of Taylor polynomial of $f(x)=e^{3x}-1$

I wonder how to compute the limit of error term in Taylor polynomial of exponential functions (i.e. $f(x)=e^{3x}-1$). First, I found the Taylor polynomial of $f(x)$, which is $$T_n ...
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0answers
13 views

looking for approximation/expansion of $f(t)=a(t)/\sqrt{b(t) + \epsilon(t)}$ with $\epsilon(t) << b(t)$

I have the following function $ f(t) = \frac{a(t)}{\sqrt{b(t) + \epsilon(t)}} $ defined for $t\geq 0$. I know that $a(t) > 0$, $b(t) > 0$, $\epsilon(t) \geq 0$ and $\epsilon(t) << b(t) $ ...
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1answer
25 views

Calculating Maclaurin series

I need to calculate $\displaystyle \lim_{x \to 0} \sin(x)\cdot \cot(\tan x)$ using Maclaurin series I took the usual approach but did not get anything of help
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1answer
28 views

Seeking possibility of more elementary means of evaluating an improper integral.

It can be shown that $\int_0^\infty -\log{(1-e^{-x})}=\zeta(2)$ by expanding out the integral as $\log(1-z)$, exchanging summation and integration, then summing up the integrals. I am wondering if ...
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2answers
150 views

How to solve $\sin (x) + \sin( x+y) =y$?

A friend of mine asked how to lower and raise a constant line on the x-y axis as a function of sines and cosines. That is where I found that $$\sin(x) + \sin(x+y) = y $$ would do the trick if I found ...
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3answers
2k views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
3
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1answer
545 views

How to use Chebyshev Polynomials to approximate sin(x) and cos(x) within the interval [−π,π]? [closed]

I have approximated sin(x) and cos (x) using the Taylor Series (Maclaurin Series) with the following results How can I use Chebyshev Polynomials to approximate sin(x) and cos(x) within the ...
1
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1answer
45 views

Taylor Expansion of $e^{itx}$, Expectation

the Taylor-expansion of $e^{itx}$ is $$1+itx+(itx)^2 / 2! + \cdots.$$ My question: Why can one write $1+itx+o(t)$ for the sum I sated above? $o(t)$ would mean that $(itx)^2 / 2! + \cdots$ would ...
3
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1answer
53 views

Closed form for $n$-th derivative of $\sqrt{f(x)}$ for general $f(x)$

Let's assume we have an inifinitely differentiable real valued function $f(x)$, and we have a closed form expression for all its derivatives. Is it then possible to find a closed form for the $n$-th ...
2
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1answer
35 views

Converge properties of Taylor Series expansion of complex function

I need to find the convergence properties of the Taylor Expansion of $$f(z)=\frac{z}{z-1}$$ I found the Taylor Series: $$\sum_{j=1}^\infty \frac{(-1)^{j+1}(z-i)^{j-1}}{(i-1)^j}$$ Then I used the ...
2
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0answers
49 views

Solving for a $v$ in $\sum a_i e^{b_i (z^2+d_i) + c_i v}$

I have an equation in complex domain, $$P(e^u,e^v)=\sum_{i=1}^{N} a_i e^{b_i u + c_i v}=0 \;\;\;\text{(A)}$$ and by redefining, at the roots (I'm only showing work for one root), the first ...
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1answer
39 views

Taylor series expansion at Infinity

I have the following function: $$f(x) = (\frac{x!}{\sqrt[]{\pi}} * (\frac{x}{e})^{-x})^6$$ I want to get the following result using Taylor expansion at Infinity: $$8 x^3+4 ...
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0answers
9 views

$f(x)-T_{f,x_0,n}=o(x-x_0)^{n} \implies f(x) \sim T_{f,x_0,n}$?

I'm not sure about this. Consider a function $f$ and it's $n$ degree Taylor Polynomial in $x_0$ $T_{f,x_0,n}$. Considered the remainder function $R(x)=f(x)-T_{f,x_0,n}=o(x-x_0)^{n}$ Can I ...
1
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7answers
76 views

How to find the Maclaurin series for $(\cos x)^6$ using the Maclaurin series for $\cos x$?

Find the Maclaurin series for $(\cos x)^6$ using the Maclaurin series for $\cos x$ for the terms up till $x^4$. Here is what I've worked out: Let $f(x) = \cos x,\ g(x) = (\cos x)^6$. $$g(x) = ...
2
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2answers
36 views

How do I calculate the error bound for a Maclaurin series?

How many terms of the Maclaurin series of $f (x) = \ln(1 + x)$ are needed to compute $\ln(1.2)$ with an error of at most $0.0001$?
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4answers
53 views

Taylor series about 0 for $\frac{x^{2}}{e^{2x}}$

I got the following question: The Taylor series about 0 for $e^{x}$ is: $ e^{x} = 1 +x + \dfrac{1}{2!}x^{2} + \dfrac{1}{3!}x^3 + \dfrac{1}{4!} x^4 + \dots \qquad \text{for $x \in \mathbb{R}$} $ And ...
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4answers
48 views

Let $f(x)=\sum_{n=0}^{\infty}a_nx^n$ be convergent on $(-R,R),$ with $R>0.$ Prove that, if $f(x)=0$, $a_n=0$.

I'm stuck on a solution that our teacher gave to us. This is the exercise: Let $f(x)=\sum_{n=0}^{\infty}a_nx^n$ be convergent on $(-R,R),$ with $R>0.$ Suppose that $f(x)=0$ for all ...
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2answers
50 views

Why is $(1+\frac{3}{n})^{-1}=(1-\frac{3}{n}+\frac{9}{n^2}+o(\frac{1}{n^2}))$ and how to get around the Taylor expansion?

Let be $(u_n)$ a real sequence such that $u_0>0$ and that $\forall n \in \mathbb{R}$: $$\frac{u_{n+1}}{u_n}=\frac{n+1}{n+3}$$ Let be $(v_n)$ a real sequence such that $\forall n \in ...
0
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1answer
16 views

Taylor Series from General Finite Difference Scheme

"For a 3-point stencil $[x_{i-1},x_{i+1}]$, we can write a generic expression as $\frac{\partial u}{\partial x}|_{x_i}=au_{i-1}+bu_i+cu_{i+1} + O(h^m)\qquad (1)$ where a,b, and c are unknowns to be ...
4
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1answer
174 views

Prove the identity $\cosh(2x)=\cosh^2(x)+\sinh^2(x)$ using the Cauchy product. [closed]

Prove the identity $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ using the Cauchy product and the Taylor series expansions of $\cosh(x)$ and $\sinh(x)$. The relations involving the exponential function are ...
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4answers
51 views

Remainder term for Maclaurin's $\sin x$ expansion

We know that for the Maclaurin's series $$\sum_{k=0}^{n}\frac{ f^{k}(0) }{(n+1)!}x^{k}$$ the remainder term is given by the following formula: $$R_{n} = \frac{\left | f^{(n+1)}(z) \right |x^{n+1} ...
4
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2answers
162 views

Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around ...
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0answers
10 views

Inequality for a multivarialbe function?

For fixed $y\in \mathbb{C}^m$ and let $f$ be a fuction defined on $\mathbb{C}^m\times \mathbb{C}^m$ such taht $f(0,y)=1$ and $$\frac{\partial^n} {\partial x^n}f(x,y)=(i)^my^n{}f(x,y)$$ which means ...
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2answers
31 views

Evaluate the function $f(x)=\frac{x^2+3e^x}{e^{2x}}$ using Maclaurin series

$$x^2+3e^x=x^2+3\sum\limits_{n=0}^{\infty}\frac{x^n}{n!}+O(x^n)=x^2+3\left(1+x+\frac{x^2}{2}+O\left(x^2\right)\right)$$ ...
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2answers
70 views

Show this assertion

I am stuck on the following task: Fix a vector $y \in C^n$ and let $f(x,y)$ be a function defined on $C^n \times C^n$ such that $f(0,y)=1$ and its $n$th partial derivative on $x$ satisfies ...
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2answers
100 views

Ordinary generating function of powers of 2

Is there a good closed form expression for the generating function of the formal power series $$ A(z) := \sum_{n=0}^\infty z^{2^n} = z + z^2 + z^4 + z^8 + z^{16} + \cdots. $$ Is there a tractable way ...
0
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1answer
47 views

Solving $(1-x^2)y''-2xy'+a(a+1)y=0$

I need to find an even solution and an odd solution to the ODE $(1-x^2)y''-2xy'+a(a+1)y=0$ using a power series around $x=0$. I suspect I should use Frobenius, but not sure how to bring it to the ...
3
votes
3answers
65 views

Maclaurin Series Representation for $f(z)=\frac{z}{z^4+9}$

I need help finding the Maclaurin series representation for $$f(z)=\frac{z}{z^4+9}$$ I first tried to factorize $z^4+9$, but am I missing something? I could not figure out how to factorize this. Is ...
4
votes
2answers
48 views

How to compute $\lim_{x \to 0} (\frac{x^5 e^{-1/x^2}+x/2 - \sin(x/2))}{x^3})$?

I have a problem with this limit. I have no idea where is the problem. Can you correct my mistake? Thanks $$\lim\limits_{x \to 0} \left(\frac{x^5 e^\frac{-1}{x^2}+\frac{x}{2} - ...
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0answers
22 views

Connection between properties of taylor series and the function

Assuming I have a function $f(x)$ which at least for some $-R<x<R$ can be expanded in taylor series $$ f(x) = \sum_{n=0}^{\infty}c_n \frac{x^n}{n!} $$ are there any known connections ...
2
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2answers
27 views

Properties of the remainder function for Taylor polynomials

Considered $f$ differentiable at least $n$ times in $x_0$ and $P_{n,x_0}(x)$ the $n$ degree Taylor polynomial in $x_0$. Defined the Remainder function $R(x)= f(x)-P_{n,x_0}(x)$ I can't understand ...
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0answers
16 views

Edgeworth expansion of the sum of inid random variables?

This question relates to the asymptotic expansion for the distribution of sum of random variables using moments. Edgeworth expansion can be applied when the variables are independent and identically ...
8
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3answers
78 views

Prove that $\cosh^{-1}(1+x)=\sqrt{2x}(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+…)$

How can we prove the series expansion of $$\cosh^{-1}(1+x)=\sqrt{2x}\left(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+...\right)$$ I know the formula for ...
0
votes
1answer
44 views

How to show that $\frac{159999}{80000} +\frac{1}{100e^2} <\ln(e^2+\frac{1}{100} ) < 2+ \frac{1}{100e^2}$

I'm trying to show that $\frac{159999}{80000} +\frac{1}{100e^2} < \ln(e^2+\frac{1}{100} ) < 2+ \frac{1}{100e^2}$. I know I should do something with the first order taylor polynomial of $\ln(x)$ ...
1
vote
1answer
26 views

Finite expansion of this function

I had this result as finite expansion of this function $$\frac{1}{(1-x)^2}$$ to order n in neighborhood of 0: $$1+\sum_{i=1}^{n}{(x^i.(i+1)) }+0(x^n)$$ (where x tends to 0)is it true? And if yes ...
2
votes
1answer
64 views

How come $\frac{1}{\cos x} = 1 + \frac{x^{2}}{2} + o(x^{2})$ as $x \to 0$?

Since $$\cos x = 1 - \frac{x^{2}}{2} + o(x^{3})$$ as $x \to 0,$ we have $$\frac{1}{\cos x} = \frac{1}{1-\frac{x^{2}}{2} + o(x^{3})} = 1 + \frac{x^{2}}{2} + o(x^{3}) + o(\frac{-x^{2}}{2} + ...
2
votes
0answers
24 views

Relation between representation of a number in an integer base and Fourier series representation of a periodic signal

I am not a Mathematician - am just a software developer though I did some "Math" back in the day as part of my undergrad studies millions of years ago. Recently I had to revisit Fourier analysis of ...
3
votes
0answers
239 views

$4$- vector Taylor expansion, sign confusion

I've been presented with a function expansion which I'm told is correct but I can't figure out where the sign in the second term might be coming from. $$ e^{i\alpha(x_\mu + \epsilon \, n_\mu)} = ...
0
votes
0answers
22 views

External and internal multipole expansion for axisymmetric potential - the region of convergence.

Say, we have a system of electrodes exhibiting symmetry around a certain axis. We know the explicit expression for the potential on the axis $\phi (z)$. We want to find the potential for any point in ...
1
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1answer
66 views

Prove (or disprove) that $ \sum_{n=1}^\infty \frac{4(-1)^n}{1-4n^2} x^n = \frac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} - 2 $ for $ 0<x\leq1$

Just like title said, for $ 0 <x\leq1 $, prove/disprove: $$ \displaystyle \sum_{n=1}^\infty \dfrac{4(-1)^n}{1-4n^2} \cdot x^n \stackrel{?}{=} \dfrac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} - 2 $$ I ...