Let $\mathbb{K}[x]$ the ring of polynomials in x with coefficients in $\mathbb{K}$. Let $$V_n = \left [nx^n + (n-1)x^{n-1} + \ldots + 1 \right ] $$
Show that $$\mathbb{K}[x] = \sum\limits_{n=1}^{\infty} V_n $$
Well, let me tell you what I've already done. It's trivial to prove the inclusion $ \sum\limits_{n=1}^{\infty} V_n \subseteq \mathbb{K}[x]$. But I've got problems with the other inclusion. Actually, I was wondering if that inclusion really holds, because if I consider the sum of all these sets, what I think I'll have, is the polynomials of the form $$\sum k_i \cdot n \cdot x^{n} $$ In other words, I think the coefficients will be multiples of $n$. Therefore, that wouldn't be the whole ring of polynomials.
What type of object is $\Bbb{K}$? If, for example, $\Bbb{K}$ is a field of characteristic zero, then $n$ is a unit, so "multiples of $n$" actually includes all elements of $\Bbb{K}$.