I have the following problem : For a given positive real $A>1$ $$A = \sum_{i=1}^{\infty} \frac{i}{a_i}$$ $$1 = \sum_{i=1}^{\infty} \frac{1}{a_i}$$ $$a_{n+1}>a_{n}$$ where $a_i$ is a strictly increasing sequence of strict positive integers. How to find $a_i$ for a given closed form $A$, in particu...

Show that the differential equation $\frac{dy}{dx}=\sqrt y,y(0)=0$ has non-unique solution. To solve this problem I used Lipschitz theorem. My solution goes like this: Let $f$ be continuous in an unbounded domain $$D:a\leq x\leq b,-\infty\leq y\leq \infty.$$ Let $f$ satisfy a Lipschitz condition...

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of mathematics ranging from exercises in freshman classes to constructing useful counterexamples for s...