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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Solve the following ODE using a Maclaurin expansion of the non-linear terms

Find two proper series solutions about the ordinary point $x=0$ of $$y''+e^xy'-y=0.$$ My proposed solution: Note that $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}.$ Assume there exists a power series ...
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1answer
21 views

Taylor and Macluarin series deriving

Hi to everyone Here i am studying Taylor series. $$f(x)=c_0 + c_1 (x-a) + c_2 (x-a)^2+ ...$$ $$ f(x)= f(a) + \frac{df(a)/dx}{1!}(x-a)^1 + \frac{d^2f(a)/dx^2}{2!}(x-a)^2 ...$$ Well my problem is ...
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15 views

Estimate an Taylor approximation II

i am doing some exercise for my numerical analysis course. And i found myself wondering if the following argument is legal. The context of this exercise is the smoothend newton algorithm, especially ...
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1answer
20 views

Laurent series about singular point for: $\left( \frac{1+x}{1-ax}\right)^{\frac{1}{3}}$

$\left( \frac{1+x}{1-ax}\right)^{\frac{1}{3}}$ I wish to find the Laurent series about the singular point $x=1/a$. I can find an expansion for the left side ($x=0$) and the right side ($x \rightarrow ...
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1answer
22 views

Taylor Series of a composition of functions

I have to find a Maclaurin series of the following function: $y = D\sin(C\arctan(Bx - E(Bx - \arctan(Bx)))) + Sh$ I wasn't able to find it by hand. Thanks in advance!
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1answer
40 views

Series expansion of infinite series raised to the $n$th power

So I know there is a well-known straightforward way to expand something like $$(a+b)^n$$ and that there are formulas which allow us to expand trinomials and multinomials in general. My question is, ...
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10 views

The remainder of taylor approximation, lagrange form of the remainder. The idea

We know that the formula for the remainder of taylor approximation is: $$R_n(x) = \frac{f(z)^{n+1} *(x-a)^{n+1}}{(n+1)!}$$ But also we have the formula: $$R_n(x) = \frac{M *(x-a)^{n+1}}{(n+1)!}$$ ...
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2answers
46 views

Find the Maclaurin series for $\cos^2(x)$

I am given this as a hint: $\cos^2(x) = \frac{1 + \cos(2x)}{2} \\$ I am not really sure how to start this one, would it just be the regular Maclaurin series squared. For example: $ (\sum_{n=0}^\...
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1answer
17 views

Relation between coefficients of two different power series.

Let $$f(z) = \sum_{n\geq 0} = a_nz^n, a_n\in\Bbb{C}$$ has a radius of convergence $\rho$. Then we can write $f(z) = \sum_{n\geq 0} b_n (z-\frac{\rho}{2})^n$ for $\{z: |z-\dfrac{\rho}{2}|<\dfrac{\...
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1answer
102 views

Approximate roots of nonlinear equation (non-integer polynomial)

In case of pulsating bubble arising from underwater explosion, bubble radius satisfies the following equation. $x^3\dot{x}^{2} + x^3 + \frac{k}{x^{3(\gamma-1)}} = 1$ The minimum and maximum bubble ...
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24 views

Mclaurin series and n-th derivative

(1) Find the general formula of the McLaurin series of $ f(x) = arctan((x^3)/2)/x^3\ $ (2) Evaluate the 18-th derivative of f(x) (3) Evaluate lim to infinity of f(x) By general formula do we just ...
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1answer
40 views

Taylor series doesn't seem to have a pattern?

My teacher gave us a study guide to work on, and one of the problems doesn't seem to come out right. The directions are to "find the Taylor series of $f(x)=x^5-3x^4+x^3+2x-1$ for $a=1$. I calculated ...
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1answer
32 views

Taylor series third order approximation

There has been this question that had been bothering me for a while and I could not find a satisfying answer on the internet or any of the books even though it is not very complex. i) Its because if ...
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1answer
41 views

Does multiplying Taylor series by an integer change the interval of validity.

If I have a Taylor series for example, $\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \ldots, \qquad \text{valid for $-1<x<1$} $ and I multiply the series by some integer, let's say $5$, in ...
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1answer
32 views

If the derivative is written as shifts, can you relate it to the laplace/fourier tranform?

I was wondering if there is a way to write the derivative as an exponential? This might sound crazy at first, but I recently came across this formula for the Taylor expansion in three dimensions: $$\...
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1answer
18 views

Taylor expansion of $f(x(t),y(t))$ around the point $(x_0,y_0)$.

My main question is basically whether the fact that both inputs depend on $t$ is an issue? Because if $x$ changes then $t$ must have changed and thus $y$ is likely to have changed. So would we need ...
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2answers
54 views

How do I show that $\ln(2) = \sum \limits_{n=1}^\infty \frac{1}{n2^n}$?

My task is this: Show that $$\ln(2) = \sum \limits_{n=1}^\infty \frac{1}{n2^n}$$ My work so far: If we approximate $\ln(x)$ around $x = 1$, we get: $\ln(x) = (x-1) - \frac{(x-1)^2}{2} + \frac{(x-...
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22 views

Solving limit of Integral through Taylor

Let $u:U\rightarrow \mathbb{R}$ ($U\subseteq \mathbb{R}^3$) be twice continously differentiable. Evaluate the limit: $$\lim_{r\to 0^+} \frac1{r^2} \Bigg( \frac1{4\pi} \iint\limits_{\xi^2+\eta^2+\...
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2answers
35 views

Can we Relate Radius of Convergence of Taylor Series and Asymptotic Rate of Growth?

I still need to be disabused of the belief that there is some simple connection between the finiteness of the radius of convergence and the asymptotic rate of growth. 1. Can we develop any ...
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0answers
57 views

Taylor series question help!

This question is on a past paper for my exam but no model solutions have been provided and I'm worried I'm doing completely the wrong thing, Consider two functions represented by Taylor (MacLaurin) ...
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1answer
19 views

Could someone please confirm my answer this Maclaurin series??

Find three nonzero terms of the Maclaurin series of the function $f(x)={3/5} tan5x/x$ Using the maclaurin series i found them to be.. $3/5+x^2/25+2x^4/25$ Is this correct? If not what is the ...
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14 views

Determine the Lagrange Residual of $\ln(\frac{1-x}{1+x})$

Show, for $x_0=0$, that $\ln(\frac{1-x}{1+x})=-2\big[x+\frac{x^3}{3}+\dots+\frac{x^{2n-1}}{2n-1}+R_{2n}(f,0)(x)\big]$, with $$R_{2n}(f,0)(x)=-\frac{x^{2n+1}}{2n+1}\bigg(\frac{1}{(1+\theta x)^{2n+1}}+\...
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0answers
380 views

Fractional Derivative of a Taylor Series?

I have a function defined only by it's taylor series: $$f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!}x^k$$ Obviously, integer derivatives can be defined as $$\frac{d^n}{dx^n} f(x) = \sum_{k=0}^\...
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41 views

Consider the function $f(x) = e^{x^2}\ln(1+x)$ for $0 < x < 1$

So I was able to do the first half of this problem (part a), which was: $$e^{x^2}\ln(1+x) \approx x - \frac{x^2}{2} + \frac{4x^3}{3}$$ but I'm confused what my next step should be, for solving (part ...
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1answer
39 views

Factorization of Taylor series.

I know that for a (finite) polynomial $P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_0$ whose zeros are $x_1, x_2, \ldots, x_n$, then we can factorize it as $$P(x) = a_n(x - x_1)(x - x_2) \cdots (x -...
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2answers
47 views

Find a function f so that Taylor expansion is always accurate to this degree

Find a function $f$ from R to N such that with $T$ be the Taylor expansion of $\sin(x)$ around $0$. $ | \sin (x) - T_{f(x)}x$| $\leq 1$ The hint is to use $n! \leq 3 \sqrt{n} {(\frac{n}{e})}^n$
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1answer
20 views

Is Every (Real) Analytic Function (with Non-Degenerate MacLaurin Series) Asymptotically Greater Than any Polynomial?

Question: Given a function $f: \mathbb{R} \to \mathbb{R}$ such that the MacLaurin series exists and equals the function for every $x \in \mathbb{R}$, and such that for all $n \ge n_0$, $n_0$ some ...
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37 views

Taylor series Lagrange Remainder explanation

So, given a Taylor series: $$f(x)=f(x_0)+f'(x_0)(x-x_0)+f''(x_0)\frac{(x-x_0)^2}{2!}+\cdot\cdot\cdot+f^{(n)}(x_0)\frac{(x-x_0)^n}{n!}+R_n$$ The error $R_n$ is given by: $$R_n=\frac{f^{(n+1)}(\xi)}{(n+...
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27 views

Could some confirm my answer for this limit using taylor series?

$\lim_{x→0}$ $\dfrac{x^2}{x\sqrt{1+x} −\ln(1+x)}= ?$ I got $-2$. Is this correct if not what is the answer so i can find out where i went wrong. Thanks in advance
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2answers
59 views

Proving divergence of a series via Taylor Expansion [closed]

I would like to prove using Taylor expansion that the series $\sum\left(\sqrt{1+\frac{(-1)^n}{\sqrt{n}}}-1\right)$ is divergent for $n\geq 1$. What is the expansion to prove it ? Thanks
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Taylor expansion in proof of weak maximum principle

Picture below is part of proof of weak maximum principle. On the red line ,I don't know how to use the Taylor expansion to get $-u''(x_0) \le 0$. I think the Taylor expansion of $u(x)$ at $x_0$ is $$ ...
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3answers
40 views

Complex analysis: Using Taylor expansion to show $|c_n| ≤ \frac{1}{r^n}\sup_{z∈C_r(0)}|f(z)|$

Consider the function $f$ is defined through the power series $$f(z) := c_0 + \sum_{n=1}^\infty c_nz^n$$ and assume that the series on the right has a radius of convergence $R > 0$. Show that $$|...
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19 views

Taylor series roots at infinity

I started thinking about this after this MathSE thread. Take a sequence of Taylor polynomials $f_n$ that converge to $f$. Does $f_n$ always have a growing number or roots in $\mathbb{C}$ which grow ...
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32 views

Limit calculate using Maclaurin series

I need help to calculate this limit using Maclaurin series: $\lim_{x\to \infty}((x^3-x^2+\frac{2}{x})e^{\frac{1}{x}}-\sqrt{x^3+x^6})$ I don't know from where to start. I think I need to to write ...
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3answers
87 views

Show that $\sin(x) > \ln(x+1)$ for any $x \in (0,1)$

Show that $\sin(x) > \ln(x+1)$ when $x \in (0,1)$. I'm expected to use the maclaurin series (taylor series when a=0) So if i understand it correctly I need to show that: $$\sin(x) = \lim\...
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1answer
23 views

Polynomial approximation of a limit

I am supposed to find the Taylor polynomial $P_2(x;1)$ for the exponent function $f(x)=e^x$ and use it in conjunction with Taylor's theorem to evaluate the following limit: $$\lim_{x\rightarrow1} \...
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4answers
107 views

prove: $\lim\limits_{x \rightarrow a} \frac{f^2(x)- g^2(x)}{(f(x) -f(a))^2} = 1$

$f(x)$ and $g(x)$ both differentiable twice at $x = a$ and we know that $f''(a) =g''(a)+f(a)$, $f(a) = g(a) = f'(a) = g'(a) \not = 0$ (we don't know if $f(x)$ and $g(x)$ are differentiable anywhere ...
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1answer
31 views

$\lim_{x \to 0}\frac{(x^2 \times 2^x \times (\log 2)^2) - (2^x - 1)^2}{(2^x - 1)^2(x^2 \times \log 2)} = ?$

$$\lim_{x \to 0}\frac{(x^2 \times 2^x \times (\log 2)^2) - (2^x - 1)^2}{(2^x - 1)^2(x^2 \times \log 2)}$$ I tried this by using the Taylor series $2^x = 1 + x\log 2 + \frac{x^2}{2!}(\log2)^2 + \dots$....
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1answer
27 views

Mclaurin series

I need to use Mclaurin series in order to show that for every $x\in$ $(0, 1)$ $\sin(x)>\ln(1+x)$ I don't know from where to start, I think I should define $f(x)=\sin(x)-\ln(1+x)$ and then to ...
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45 views

Approximating functions using Taylor polynomials

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide any explanations. I literally have no idea how to ...
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2answers
60 views

What's wrong with my Taylor -Maclaurin- Series? $e^{x^2+x}$

Here's what I have: We know: $$e^x = 1 + x + \frac{1}{2!}x^2+\frac{1}{3!}x^3 +\frac{1}{4!}x^4$$ Now I can calculate the Taylor Series for $e^{x^2+x}$: $$1+u+u^2+\frac{1}{2!}(x^2+x)^2+\frac{1}{3!}(...
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1answer
39 views

Integrate $\frac{\sin x^3}{x^3}$ as a power series

Today, I tried to do this by taking the MacLaurin of Sin to 4 terms, putting in $x^3$ in place of $x$, multiplying the terms by $x^{-3}$, and integrating. I came out with a sum that had $x^{6n+1}$ as ...
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244 views

Is there a way to write this recurrence relation in faster-to-program manner?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
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1answer
108 views

prove: if $f(\frac{1}{n}) = 0$ for all $n \in \mathbb{N}$ then $f(x)=0$ for all $x \in \mathbb{R}$

$f(x)$ is infinitely differentiable and $\exists L \in \mathbb{R}$ such that $|f^{(n)}(x)| \le L$ for any $n \in \mathbb{N}$. I need to prove that given the information above: if $f(\frac{1}{n}) = 0$ ...
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2answers
42 views

The value of $\lim_{n\rightarrow\infty}[(n+1)\int_{0}^{1}x^{n}$ $\ln(1+x)$ $dx]$

I evaluated it as $\lim_{n\rightarrow\infty}[x^{n+1}ln(1+x)]_{0}^{1}-\int_{0}^{1}x^{n+1}(1+x)^{-1} dx$ , which comes as $\ln (2) - \lim_{n\rightarrow\infty}\int_{0}^{1}x^{n+1}(1-x+x^{2}-x^{3}\ldots)dx ...
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1answer
32 views

How can I evaluate the limit of this function using series?

Limit as x approaches 0 of $lim_{x\rightarrow 0}\frac{1-cosx}{1+x-e^x}$. I substituted in the Taylor series of $cosx$ and $e^x$ into the function, but it's still in $\frac{0}{0}$ form, and I don't ...
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4answers
102 views

How to evaluate this limit using Taylor expansions?

I am trying to evaluate this limit: $\lim_{x \to 0} \dfrac{(\sin x -x)(\cos x -1)}{x(e^x-1)^4}$ I know that I need to use Taylor expansions for $\sin x -x$, $\cos x -1$ and $e^x-1$. I also realise ...
2
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4answers
178 views

Show $\int_0^1 \frac{\log(1-x)}{x}dx=-\frac{\pi^2}{6}$

It's claimed that $$\int_0^1 \frac{\log(1-x)}{x}dx=-\frac{\pi^2}{6}$$ by first expanding $\frac{\log(1-x)}{x}$ into a power series and then doing term-by-term integration. I want to justify this by ...
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0answers
16 views

Proof relating to the remainder term in Taylor's theorem

I'm asked to show that $\left\lvert R_n(x) \right\rvert \leq \frac{\left\lvert x \right\rvert ^n}{n!}\sup \left\lvert f^{(n)}(t)\right\rvert$ where $$R_n(x)=\frac{1}{(n-1)!}\int_0^x (x-t)^{n-1} f^{(n)}...
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1answer
349 views

How to compute the values of this function ? ( Fabius function )

How to compute the values of this function ? ( Fabius function ) It is said not to be analytic but $C^\infty$ everywhere. But I do not even know how to compute its values. Im confused. Here is the ...