# Tagged Questions

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### Notation for near optimal solution

Usually, $x^*$ is used to denote the optimal solution to a maximization problem. I need a notation to describe a solution that is not optimal but "good enough." In my case this solution is the first ...
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### Taylor series of $\sqrt{1+x}$ using sigma notation

I want help in writing Taylor series of $\sqrt{1+x}$ using sigma notation I got till $1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\ldots$ and so on. But I don't know what will come in ...
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### Vertical bar notation: $\frac{d}{dt}|_{t=0}f(a+tv)=$?

The definition of the directional derivative is $D_vf(a)$=$\left.\dfrac{d}{dt}\right|_{t=0}f(a+tv)=\lim \limits_{t\to0}\dfrac{f(a+tv)-f(a)}{t}$. However, I do not understand how this bar notation is ...
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### What does this notation involving greater than sign mean?

Basically it says that $$|x|>1,2n|a_{n-1}|,\ldots,2n|a_0|$$ Does this mean that $$|x|>\max(1,2n|a_{n-1}|,\ldots,2n|a_0|)$$ It seems to be essential when proving that polynomial whose degree ...
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### Simplification of a nested sum

I have a nested sum like so: $$\underbrace{\sum_{k_1=k_0}^{k^*} \ ... \sum_{k_n=k_{n-1}}^{k^*}}_{\text{n times}} 1\quad\ \text{with}\ \ n, k_0, k^* \in \mathbb{N},\ k^*\geq k_0$$ Is there a general, ...
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### Seperation of variables justification?

I haven't found a similar question on Math SE, but I may not have looked enough because I find it hard to believe someone hasn't already asked this. Anyways, here goes: I'm studying mathematics, but ...
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### Integrating With Respect To $x$

Suppose I have the first derivative of the function $y$, $\displaystyle \frac{dy}{dx} = g(x)$. Furthermore, suppose I want to obtain the function $y$ by integrating with respect to $x$: ...
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### Equation with the big O notation

How I can prove equality below? $$\frac{1}{1 + O(n^{-1})} = 1 + O({n^{-1}}),$$ where $n \in \mathbb{N}$ and we are considering situation when $n \to \infty$. It is clearly that it is true. But I ...
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### What is meant by this notation?

I found the following question: Let $\large f(x)=e^{x^2}$. Find and simplify $\large{f^{(3)}(x)}$. Is it asking to find the third derivative of the function? Which rule is used to find the ...
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### Notation: using dot instead of argument

Is there any difference between: $f\left(\cdot , \theta \right)$ is continuous in x for each $\theta$ $f\left(x , \theta \right)$ is continuous in x for each $\theta$
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### Partial differentials vs normal differential (notation question/clarification only)

In physics, it seems like the use of $\dfrac{dy}{dx}$ and $\dfrac{\partial y}{\partial x}$ are used somewhat interchangeably. My understanding is that, technically $\dfrac{dy}{dx}$ is only ...
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For functions of one argument, the "Newton" notation for the derivative of that function is concise and unambiguous. For example, if I want to express $$\lim_{h\to 0} \frac {f(x^2 + h) - f(x^2)}{h} ... 2answers 60 views ### What does this function mean?$$f(x) = \frac{x}{e^{x^2}}$$Differentiate f(x). How should the above function be interpreted? Is the function equivalent to: a)$$f(x) = \frac{x}{e^{x^2}} = \frac{x}{{(e^x)^2}} = ...
I'm currently Rudin's Principles of mathematical analysis, there is this definition of the "partial derivative". If $f$ maps an open set $E\subseteq R^n$ into $R^m$ and $\{e_1,...,e_n\}$ and ...