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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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 ...
2
votes
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, ...
2
votes
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
35 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
117 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 ...
0
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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 ...
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1answer
25 views

Give the Taylor series for the following $f(z)$; also, find $f^{(100)}(0)$

$$e^{3z}$$ I'm not sure how to approach this complex number problem. I know that $$1+3 x+\frac{9 x^2}{2}+\frac{9 x^3}{2}+\frac{27 x^4}{8}+\cdots$$ is true for $e^{3x}$, but how does this apply to ...
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3answers
30 views

Convert $f(x)=(\cos(x))^3$ to powers of x and find if converges.

I started out by writing the Taylor series for $x_0=0$ (Maclaurin series) of $f(x)=(\cos(x))^3$. If my calculations are correct $$f(x)=1-\frac{3x^2}{2!}+\frac{21x^4}{4!}-\frac{183x^2}{6!}+...$$ and ...
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2answers
85 views

Over an integral arising from Kepler's problem [also: generally useful integral, NOT DUPLICATE!]

This post might appear as a duplicate of the following: Over an integral arising from Kepler's problem [also: generally useful integral] So recalling quickly: $$\Phi(\epsilon) = ...
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3answers
58 views

Series expansion: $1/(1-x)^n$

What is the expansion for $(1-x)^{-n}$? Could find only the expansion upto the power of $-3$. Is there some general formula?
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1answer
18 views

Comparing a series expansion to polynomial regression

So I don't have a great background in mathematics but I have a quick and hopefully simple question for you guys. I'm a graduate student and I'm doing some polynomial regression on some thermodynamic ...
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1answer
13 views

How to use given permissible error.

If Taylor series expansion of $\cos(x)$ is restricted to only first two terms and the permissible error is$ .54×10^{-2}$, then $x$ can be at most. Should i use the permissible error as this ...
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2answers
23 views

Sufficient conditions for applying Taylor theorem

Consider a real-valued function $f:\mathbb{R}\rightarrow \mathbb{R}$. Is assuming $f(.)$ twice differentiable at $a \in \mathbb{R}$ enough to apply the Taylor Theorem stating $$ ...
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2answers
34 views

Order of $x^{1/x}-1$ as $x \to \infty$

Order of $x^{1/x}-1$ as $x \to \infty$ Answer been given: $x^{1/x}-1=O(\frac{\ln x}{x})$ My attempt is to expand the equation: $x^{1/x}=exp(\frac{\ln x}{x})$ I am unable to proceed further, any ...
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4answers
54 views

Expand $\sqrt{x(1-x)}$

I am trying to expand $\sqrt{x(1-x)}$ in order to find the order of it as $x\to0$. Using Taylor expansion to find the derivative of the equation as $x\to0$ gives each item $0$. Then how do we expand ...
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0answers
29 views

Is it possible to very roughly approximate a Taylor Series expansion? [closed]

Say I have the following equation: $$ y=-0.000001x^6+0.00001x^5+0.0001x^4+0.001x^3-0.01x^2-0.1x+1 $$ Is it ever possible to find a Taylor polynomial that is similar enough (but not exactly the ...
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0answers
14 views

Solving $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $

I would like to work out the result of $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $. Here, $t, i, N_i, m_i$ are positive integers. My effort: $$ \Pi^t_i 2 m_i \left(N_i!\right)^{m_i} \implies (2 m_1 ...
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1answer
188 views

Use Stirling's formula to estimate the location and size of the largest term of the Taylor series of $e^x$

Use Stirling's formula to estimate the location and size of the largest term of the Taylor series of $e^x$. I don't know how to start. Thanks
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1answer
39 views

Using Lagrange Remainder to find the approximation of $\sqrt(8)

I'm looking for an approximation of $\sqrt 8$ with an approximation of $10^{-4}$. It was given that $\sqrt 8 =3\sqrt\frac{8}{9}$ so I set up a general series for $ \sqrt{1+x}^\frac{1}{2} $ around zero ...
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0answers
15 views

Find Taylor series for function and use it to calculate sum of a series. [duplicate]

So i need in finding the correct Taylor series(expansion) for the following function and use it in calculating the sum of series: $$f(x)=\ln(\frac{1-4x^2}{1+4x^2})$$ ...
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2answers
50 views

MacLaurin series for $9\sec(3x)$

A question I've been given asks me to find the first 3 non-zero terms of the MacLaurin series for the function: $y = 9sec(3x)$ Looking at old questions on this forum, I think that this is supposed to ...
0
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1answer
21 views

Why $T_{\infty}\sin(x^2)$ could be written as $x^2+o(x^4)$?

I am confused about the little-oh notation in the Taylor series. As we know, $T_{\infty}\sin(x)=x - \frac{x^3}{3!}+\frac{x^5}{5!}+\cdots+(-1)^n\frac{x^{2n+1}}{(2n+1)!}+\cdots$ By substitution, I got ...
0
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3answers
82 views

What is the Taylor series of $\ln \frac{1-x^2}{1+x^2}$?

What is the Taylor series of $$\ln \frac{1-x^2}{1+x^2}$$ ? I started by evaluating the first derivatives but the more I go, the more complicated they are and I can't identify a pattern.
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3answers
36 views

Taylor series for $f(x)=\cot(x)$

Trying to expand $f(x)=\cot(x)$ to Taylor series (Maclaurin, actually). But I keep "adding up" infinities when using the formula. (Because of $\cot(0)=\infty$) Could you perhaps give me a hint on how ...
3
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2answers
59 views

Maclaurin Series of $e^{-x^2}$

The question is: Find the first 3 non-zero terms in the MacLaurin series for the function: $$y =e^{-x^2}$$ I have been told to simply substitute the $-x^2$ into the standard MacLaurin series for ...
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1answer
29 views

Looking to have my worked check on a calculus series question

I am trying to determine if the Taylor series of $f(y) = y^{-\frac{1}{3}}$ about $y=1$ converges absolutely at $y = 2$. I am calculating the Taylor series as $$f(y) = 1 + a_1 (y-1) + a_2 (y-1)^2 + ...
0
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1answer
38 views

Taylor expansion of $\cosh^2(x)$

I have a question which I'm troubling to solve. I've been given the following, a Taylor expansion of $\cosh(x)$ around $x=0$; $$\cosh(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}.$$ Now, using that ...
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0answers
25 views

LTE for the Cahn Hilliard Equation

I'm trying to Calculate the LTE of a first order finite difference (central difference) scheme for the following 1D equation: $u_t = \frac{\partial^2}{\partial x^2}(u^3 - u - \frac{\partial^2 ...
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2answers
43 views

Calculate $\lim_{x \to 0} \frac{\sin(\sin x)-x(1-x^2)^\frac{1}{3}}{x^5}$ by Taylor's theorem.

I have to calculate : $$\lim_{x \to 0} \frac{\sin(\sin x)-x(1-x^2)^\frac{1}{3}}{x^5}$$ by using Taylor's theorem. I know that :$$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...
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0answers
19 views

How to find the value of $x$ that $T_{\infty} f(x)$ convert to $f(x)$?

I am not sure whether I am right about the general solution to these questions. I think Taylor series convert to the original function only when $\lim_{n \rightarrow \infty}R_{n}f(x) = 0$ However, ...
0
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2answers
22 views

Clarification of notation in multivariate taylor expansion

I'm reading the book "Numerical Optimization" by Nocedal and Wright and on page 14 of the book they present a form of the multivariate Taylor theorem which I find to be a bit peculiar. It is stated ...
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2answers
73 views

What is a power series representation of $\frac{1-x^2}{1+x^2}$

I got so far as to rewrite to this: $(1-x^2)\cdot \frac{1}{1-(-x^2)}$ so that I can write the power series as follows: $$(1-x^2)\cdot\sum_{n=0}^\infty(-1)^n \cdot x^{2n}$$ But how can I bring the ...
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3answers
58 views

Hundredth Derivative From Taylor's polynomial for $\frac{x^2}{1+x^4}$

I'm trying to solve this problem: Find the hundredth derivative (at $x=0$) from Taylor's polynomial for $\dfrac{x^2}{1+x^4}$. I keep getting the wrong answer; can someone help? I have tried ...
0
votes
1answer
37 views

Taylor series for $\frac{1}{z-5}$ at $z=i$

I need to find Taylor series for $\frac{1}{z-5}$ at $z=i$ My attempt: $$f(i)=\frac{1}{i-5}\\ f'(i)=-\frac{1}{(i-5)^2}\\ f''(i)=\frac{2}{(i-5)^3}\\ f'''(i)=-\frac{6}{(i-5)^4}\\ \Longrightarrow ...
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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, ...
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2answers
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.
3
votes
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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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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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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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) $ ...
0
votes
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
0
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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 ...
2
votes
2answers
151 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 ...