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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2
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3answers
282 views

Find Maclaurin series for$f(x) = \frac{2x}{1-5x^3}$

I'm trying to find the Maclaurin series for $f(x) = \frac{2x}{1-5x^3}$, but my solution is different from what I know it supposed to be, which is $2x+10x^4+50x^7+250x^{10}+...$ This is my attempt: ...
1
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1answer
147 views

Derivative of a little-o remainder

If we have a $\phi: \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$, $\phi = \phi(t, \mathbf{q},\alpha)$ one-parameter group of infinitesimal transformation which is $\mathcal{C}^2$ ...
2
votes
1answer
119 views

Why do power series converge to a function symmetrically?

Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$? The selected answer to the above question says that for a a power series, the interval of convergence for the ...
1
vote
1answer
66 views

What is the exponential series representation of $x^x$?

I want to express $x^x$ in the form of an infinite series involving $qe^{sx}$ where $q$ is the $s$th coefficient of the series and $s$ is the power on $e^x$. Beyond just an answer I would like to know ...
2
votes
4answers
743 views

What exactly are the “higher order terms” (H.O.T.) in Taylor series expansion in multivariable case?

I know the Taylor series expansion in single variable case: $$ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{3!}f^{(3)}(x_0)(x-x_0)^3 + \frac{1}{4!}f^{(4)}(x_0)(x-x_0)^4 ...
0
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1answer
2k views

How to expand this taylor series and find radius of convergence

f(x)= √(1-x) at x=0 How do you find the taylor series and radius of convergence?
2
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1answer
255 views

How do you find Taylor series and radius of convergence for $\sqrt{x}$?

How do you find interval and radius of convergence of $f(x)=\sqrt{x}$ at $x=1$
2
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1answer
168 views

expression for Remainder in Taylor theorem for complex variables

I don't understand how the following summation vanishes with $j=k+1$ in wikipedia article .$$R_k(z) = \sum_{j=k+1}^\infty \frac{(z-c)^j}{2\pi i} \int_\gamma \frac{f(w)}{(w-c)^{j+1}}dw = ...
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3answers
36 views

Demonstrate that $\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6}\ge 0$

How do I prove the inequality? $$\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6} \geq 0$$ I can see that $e^e = \sum_{k=0}^{\infty} \frac{e^k}{k!} = 1 + e + \frac{e^2}{2} + \frac{e^3}{6}+\dots ...
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2answers
49 views

Show that $|x-\ln(1+x)| \leq cx^2$

Use Taylor's Theorem to show that there is some fixed constant $c>0$ such that $$|x-\ln(1+x)| \leq cx^2$$ for all $|x|<\frac{2}{3}$. My attempt: Let $f(x)=\ln(1+x)$. Then by calculating the ...
1
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1answer
463 views

Curvature via hessian in Taylor expansion

In the case of a univariate function, the smaller the second derivative in its Taylor expansion, the smaller is the curvature of the univariate function. Now, how is the curvature of the function ...
0
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2answers
91 views

Taylor series of $f(x^2)$

If you know the taylor series for $f(x)$ can you find the taylor series for $f(x^2)$ by letting $x = x^2$? The taylor series in question is $\cos(x^2)$ I know the taylor series for $\cos(x)$ is ...
0
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3answers
121 views

$k$th term of Taylor series of function $f(x) = 3x^3-2x+4$

I'm having trouble solving this question. I have all the values right but can't figure out a way to find the $k$th term. My Taylor series values up to that point are ...
1
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1answer
92 views

Question about taylor series.

let $$f(x) = \begin{cases} \frac{\cos x -1}{x^2} & \text{for } x \neq 0 \\ \\ \\ -\frac{1}{2} & \text{for } x = 0 \end{cases} $$ The Taylor series for this is $$\dfrac{1 - ...
1
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2answers
97 views

Find the sum function of $\sum_{n=0}^{\infty}\frac{n(n-2)}{n+1}x^{n-1}$

series summation: $$\sum_{n=0}^{\infty}\frac{n(n-2)}{n+1}x^{n-1}$$ where $-1 <x <1$ is there a convinient function that sums the above series? (unsure but this may be an expanded taylor ...
0
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3answers
58 views

Finding the Maclaurin series representation

$$f(x)=\frac { x }{ { (2-x) }^{ 2 } }$$ I tried finding the first derivative, the second derivative, and so on, but it just keeps getting more complicated, so I suspect I have to use binomial series. ...
0
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2answers
129 views

The coefficient of the Taylor series is given by?

what is considered the coefficient in a taylor series? and how would i solve this The coefficient of $(x-1)^3$ in the Taylor series of $ f(x) = ln x$ about $a=1$ is given by ?
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2answers
72 views

Taylor Series of $\sin 2x$ finding $f^{(n)} (a)$ where $a = 0$

ok so i get; f (x) = sin 2x f ' = 2cos 2x f '' = -4sin 2x f ''' = -8cos 2x f '''' = 16sin 2x f ''''' = 32cos 2x f (0) = 0 f '(0) = 2 f ''(0) = 0 f '''(0) = -8 f ''''(0) = 0 f '''''(0) = 32 ...
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1answer
127 views

Why do Maclaurin series approximate a function for negative domain values?

A common analogy used as an intuitive explanation for a Maclaurin series is that of a car. If you know the position, velocity, acceleration, jerk etc. of a car at time zero, you are able to predict ...
2
votes
1answer
104 views

Don't understand how to start with this assignment question

I'm working on the last problem in an assignment, and need some guidance on what to actually start by doing. The question is asking me to use taylor expansion to determine the leading error term ...
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3answers
90 views

$\frac{1}{1-x}$ series expansion

How do I know that the expression: $$\frac{1}{1-x}$$ Is equal to the infinite sum: $$-\left(\frac{1}{x}\right)-\left(\frac{1}{x}\right)^2-\left(\frac{1}{x}\right)^3-\left(\frac{1}{x}\right)^4+...$$ ...
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2answers
79 views

Basic Taylor expansions question

How can I use : $$f(x \pm h) = f(x) \pm hf^\prime(x) + \frac{h^2}{2} f^{\prime\prime}(x) + O(h^3)$$ to Prove: $$f^\prime(x)= \frac{f(x+h)-f(x-h)}{2h} + O(h^2)$$ Thanks in advance
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0answers
41 views

Inequality of Partial Taylor Series

For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k=0}^{N} \frac{x^k}{k!} ...
1
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2answers
52 views

Taylor series $ \sqrt{\frac{t}{t+1}}$

Could someone tell me how to calculate $ \sqrt{\frac{t}{t+1}}$ it should be $ \sqrt t - \frac{t^{\frac{3}{2}}}{2} +O(t^{\frac{5}{2}}) $
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2answers
1k views

Finding the 9th derivative of $\frac{\cos(5 x^2)-1}{x^3}$

How do you find the 9th derivative of $(\cos(5 x^2)-1)/x^3$ and evaluate at $x=0$ without differentiating it straightforwardly with the quotient rule? The teacher's hint is to use Maclaurin Series, ...
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4answers
3k views

MacLaurin series of $\ln(1-x^2)$

The MacLaurin series for $\ln(1 + x)$ is obtained from the series for $\frac{1}{1 + x}$ by integration. Use this and appropriate substitutions to obtain the MacLaurin series for $\ln(1-x^2)$. ...
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1answer
111 views

Taylor series expansion of $\ln(1+x)$

I want to show that $$\log(1+x) = x - \frac{x^2}{2} + R_{3} = x - \frac{x^2}{2} + \frac{x^3}{3} + R_{4}$$ with $R_{3}$ and $R_{4}$ found in the Lagrange form of the remainder in Taylor's theorem, and ...
2
votes
2answers
710 views

Finding the Taylor series of $\log x$ at $x=1$ and $2$

How do I find the Taylor series of the following functions: $f(x)=\log(x)\ (x>0)$ at the point $x=1$ $g(x)=\log(x)\ (x>0)$ at the point $x=2$ Help greatly appreciated
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0answers
49 views

Evaluating Taylor coefficient of $e^z$

According to the theory on Taylor polynomials for a complex variable, the $n$-th Taylor coefficient for $e^z$ is given by: $$ \frac{n!}{2\pi i} \int_{C} \frac{e^z}{z^{n+1}} dz. $$ How do we evaluate ...
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4answers
4k views

Maclaurin Series of $1/(1-x)$ derived from maclaurin series of $(1+x)^n$

Is there a way to derive the Maclaurin series for $\frac{1}{(1-x)}$ after finding the Maclaurin series for $(1+x)^n$ which is $\displaystyle\sum\limits_{k=0}^\infty \frac{f^k(0)}{k!}*x^k$. From ...
6
votes
3answers
869 views

Explanation of Maclaurin Series of $x^\pi$

I am reviewing Calc $2$ material and I came across a problem which asked me to explain why $x^\pi$ does not have a Taylor Series expansion around $x=0$. To me it seems that it would have an expansion ...
2
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0answers
53 views

Find a bounded function with a supporting point

Given, $g(Z)=\operatorname{tr}\phi(Z)$, where $\phi(Z)= Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right) Z$ where $Z$ is a real rectangular matrix with more rows than columns (tall and ...
0
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1answer
619 views

Deriving Taylor series for function from geometric series

Given the geometric series $\frac{1}{1-z} = \sum_{n=0}^n = 1 + z + z^2 + ...$ If there is a function $f(z)=\frac{1}{z+j}$ how would you get it's Taylor series about center z = 1? I have tried the ...
2
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0answers
64 views

How might I find all functions that have a power series whose coefficients are all 1 or -1.

Suppose I have that $f(x)= \pm 1 \pm x \pm x^2 \cdots$ How can I find $f(x)$? Specifically, I know I'm looking for all functions such that $|f^{k}(0)|=k!$, which functions are those? Which ...
3
votes
1answer
521 views

Taylor series with functions as parameters (as opposed to variables)

I'm doing my own research on the Euler-Lagrange equation and came across a proof in van Brunt's textbook "The Calculus of Variations". However, there is something I don't quite understand. Here is an ...
2
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3answers
193 views

Showing $\log(2)$ and $\log(5)$

How do I show that: $$\log(2)=\sum^\infty_{n=1}(-1)^{n+1}\frac{1}{n}$$ and that $$\log(5)=\log(3)+\sum^\infty_{n=1}(-1)^{n+1}\frac{2^n}{n3^n}$$ Thanks in advanced.
1
vote
1answer
192 views

Upper bound on the partial sum of exponential series

How can I upper-bound the following function: $$f(n;a)=\sum_{k=0}^{n-1}\frac{(n-a\sqrt{n})^k}{k!}$$ where $0<a<\sqrt{n}$ is a constant. Since it's a partial sum of exponential series, a ...
2
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3answers
118 views

Finding Maclaurin series of a given function

For function: \begin{align} f(x) = \frac{x}{1+2x} \end{align} Can be written as: \begin{align} f(x)=x\bigg(\frac{1}{1+2x}\bigg)\tag{1} \newline = x\sum_{n=0}^\infty (-1)^n2^nx^n\tag{2} \newline = ...
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3answers
85 views

Proof by using taylor series

So, everyone that took Single Variable Calculus (calc 1) should be familiar with Taylor Series. Now, I have a question: How do I show that: $$\log(2)=\sum^{\infty}_{n=1}(-1)^{n+1}\frac{1}{n}$$ and ...
2
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0answers
2k views

power series of arcsin(x) centered at x = 0

I am trying to prove that the Taylor expansion of $\arcsin(x) = \sum\limits_{n=0}^\infty \cfrac{(2n!)x^{2n+1}}{(2^nn!)^2(2n+1)}$. Sorry about the notation, I'm not sure what syntax to use. S stands ...
3
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1answer
90 views

Intuition behind 2nd order approximation, help please

I know how to apply the formula for Taylor Expansions. But what I want to understand is the intuition. Let me explain with the following example: If $y=x^5$ its 1st derivative is $5x^4$ and its 2nd ...
0
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2answers
82 views

Taylor's theorem for $|x|<1$ for $\sqrt{1+x}$?

I'm trying to do a Taylor expansion on $\sqrt{1+x}$ for $|x|<1$ but I'm not sure how to proceed after finding the derivatives. I'd understand how to do it if it were centered at $a$, but the ...
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0answers
85 views

Taylors Inequality to evaluate $f(x) = x\sin(x)$ when $a = 0$ and $-1\le x\le1$

Trying to calculate the error of this function when you use a Taylor expansion to degree 4. I keep getting $.039$ when the answer in the back of the book is $.042$. I take the fifth derivative of ...
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0answers
70 views

Why the ith coefficient of $|\lambda I-A|$ is the sum of all $i$-th order principle minors of $A$?

I come across a theorem that $f(\lambda )=|\lambda I-A|$, which equals to $\lambda ^{n}-a_{1}\lambda ^{n-1}+\alpha _{2}\lambda ^{n-2}-...(-1)^{n}a_{n}$ where $a_{i}$ is the sums of all ith order ...
2
votes
1answer
300 views

Radius of convergence of Maclaurin series for $\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$

What is the radius of convergence of the Taylor series about $z=0$ for $h(z)=\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$? Here's a plot ...
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2answers
46 views

Lagrange remainder to approximate $3^{2.1}$ less than 0.1

How do I solve this problem: Use the appropriate Taylor polynomial $P_n(x,c)$ to estimate $3^{2.1}$ with error less than $0.1$, given $\ln 3$ is about $1.099$. I understand that the remainder ...
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vote
2answers
100 views

Question about Maclaurin Series for $\cos x$

I understand how to get the proper maclaurin series representation for $\cos x$, but I'm having trouble understanding the following part conceptually: I get $\cos x$ as $\sum_{n=0}^\infty ...
1
vote
1answer
172 views

Find taylor polynomial that approximates e^x with accuracy at least 1.

Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$. I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
8
votes
0answers
150 views

Function $f(x)$, such that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$

Consider a function $f(x)$. Define Taylor series $\sum_{n=0}^{\infty} f(n) x^n$. Is there such a function, other than constant $0$, that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$? The Taylor series of ...
10
votes
1answer
335 views

Old versus New enunciation of Taylor's Theorem.

I am studying from Spivak' Calculus, and he states Taylor's Theorem as follows: THEOREM Let $f',\cdots,f^{(n+1)}$ be defined on $[a,x]$ and let $R_{n,a}(x)$ be defined by ...