# Tagged Questions

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### Constructing a sequence of function with bounded derivative

Let $f:\mathbb R\mapsto\mathbb R$ be a smooth function and analytic at $x=0$. I wish to find a sequence of functions $\{f_n\}$ such that $\{f_n(x)\}$ is convergent to $f(x)$ for all $x$ and $f'''_n$ ...
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### Why cant we do substitution in differentiation but is ok in taylor series?

I have the same question 10 year ago when i was studying high school. I dont understand it and i give up the math. 10 year ago, i need to work with calculus during work and this question come to find ...
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### Series representation of function with fractions, logarithms, squares and cosines.

I'm looking for a series representation for $$\dfrac x{x^2+(\log \cos x)^2}$$ Where $x\in(0,\pi/2)$ Note: Both finite and infinite series are accepted. I have tried taylor series, but it requires ...
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### Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
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### Question about a solution to a problem involving Taylor's theorem and local minimum

I've been studying "Berkeley Problems in Mathematics, Souza, Silva" and I came across this problem: Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $C^{\infty}$ function. Assume that $f(x)$ has a ...
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### Taylor series expansion example

I was reading an article and there was a snippet with a taylor series expansion as shown below: My question is, should (11) read as $F(xA+h)+(xΔA+Δh)\frac{\partial}{\partial x}F(xA+h)$ instead of ...
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### 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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### 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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### $\frac{\mathrm d^n}{\mathrm d x^n} e^{-\frac {1}{x^2}} = 0$ at $x=0$ [duplicate]

This is an exercise from David Brannan's Mathematical Analysis. I've proved parts (a) - (c) but need help with Part (d). Any guidance appreciated. EDIT I have solved it, by induction using the ...
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### Taylor expansion of an integral

I am interested in the Taylor series expansion around $t=0$ of the following expression: $$I(t)=\int_{0}^{\infty}e^{-x^2}\log\left(e^{-(x-t)^2}+e^{-(x+t)^2}\right)dx$$ Normally, I would proceed by ...
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### Application for mean value theorem

$f(x)$ is three-times differentiable on $[a,b]$, how to show that there is $\varepsilon\in(a,b)$ such that $$f(b)=f(a)+\cfrac{1}{2}(b-a)[f'(a)+f'(b)]-\cfrac{1}{12}(b-a)^3f'''(\varepsilon)$$
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### How to expand $x \sqrt{4 - x}$ to Maclaurin series?

Here is the task: using standard expansions, expand $f(x) = x \sqrt{4-x}$ to Maclaurin's series. I calculated derivatives up to $f^{(5)}(x)$, and got some results. Fortunately, in Maclaurin's ...
### Little question about finding a MacLaurin expansion for $f(x)=\frac{x^2}{1-x}$
First off all, I am sorry if my english is not perfect. I need help again for this exercise: Find Maclaurin series expansion for $f(x)=\frac{x^2}{1-x}$. That's what I did: ...