We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if: For every sequence $(t’_n)_{n\geq0}$, there’s a sub-sequence $(t_n)_{n\geq0}$ such that $f(t+t_n)$ converges uniformly in $\mathbb{R}$ to a function $g(t)$ i.e. $$\sup_{t\in \mathbb{R}}|f(t+t_n)-g(t)|\to 0, \ \ when \ \ n\to +\infty.$$ This class of functions was proved to be the […]

I need somebody to explain me why $$\frac{d}{dx}(y^2)=2y\frac{dy}{dx}$$ I don’t get it. I’d say the left hand side is 0, because there is no change in $y^2$ when x changes slightly.

Need some help finding this limit: $$\lim_{x\rightarrow1}\frac{\frac{1}{\sqrt{x}}-1}{x-1}$$ Here is what I have so far: $$\lim_{x\rightarrow1}\dfrac{\dfrac{1-\sqrt{x}}{\sqrt{x}}}{x-1}$$ $$\lim_{x\rightarrow1}\dfrac{1-\sqrt{x}}{\sqrt{x}}\cdot\dfrac{1}{x-1}$$ $$\lim_{x\rightarrow1}\dfrac{1-\sqrt{x}}{x\sqrt{x}-\sqrt{x}}$$ At this point I keep getting results I don’t like, I have tried multiplying by the conjugate but I keep getting denominators of $0$. What am I missing here? Thanks

prove that the rose $r=\cos(n\theta)$ (in the polar plane) has $2n$ “petals” when $n$ is even. How can I start this demonstration? I would appreciate your help

Given a succession of real numbers $\{a_n\}_n$ with $n \in \mathbb{N}$, I would like to prove, or find a counterexample, of the fact that if $|a_n|<1$ from a $n$ on then, then $\begin{aligned} \sum_{n=0}^{\infty} (a_n)^n \end{aligned}$ converges. Someone could give me a hand? Thank you!

I have found the following problem in “Introduction to Analysis” by Rosenlicht. I am not sure if my solution is correct and I highlighted my uncertainties. First we have to show that the set $$S=\Bigg\{\sum_{i=1}^N d(f(x_{i-1}),f(x_i)):x_0,x_1,\ldots , x_N \mbox{ is a partition of } [a,b]\Bigg\}$$ is bounded from above and that the sequence of the […]

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$. The Mean Value Theorem states: a function $f$ which is continuous on the closed interval $[a, b] $ $^{\textbf{(1)}}$ and differentiable on the open interval $(a, b)$ $^\textbf{(2)}$ has at least one […]

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution

If $\vartheta_{2}(q)$ is jacobi’s theta function, what is the limit $$\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$$ for the nome $q$. I would like to know whether the limit exists or not. If it does, please let me know and provide it’s evaluation

This problem is from calculus-2 course. The basic knowledge includes integral test and $p$-series test. Find an $N$ so that $$\sum_{n=1}^\infty {1\over n^4}$$ is between $$\sum_{n=1}^N {1\over n^4}$$ and $$\sum_{n=1}^N {1\over n^4} + 0.005$$ The Answer is $N=5$. How to solve it? Thanks!

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