Sum of Divergent and Convergent Series

Can an idempotent matrix be complex? How does ring theory connects with analysis? Finding $\sum_{k=0}^{n-1}\frac{\alpha_k}{2-\alpha_k}$, where $\alpha_k$ are the $n$-th roots of unity Motivation for the rigour of real analysis A smooth function $f(x)$ has a unique local and global minimum. What happens to its location as $f(x)$ varies smoothly in time? Proof that ${2x\over 2x-1}={A(x)\over A(x-1)}$ for every integer $x\ge2$, where $A(x)=\sum\limits_{n=0}^\infty\prod\limits_{k=0}^n\frac{x-k}{x+k}$ Why does limit $\lim_{(x,y) \to (0,0)}\frac{2x}{x^2+x+y^2}$ not exist? Show that B is a nonsingular matrix (not that obvious). If a linear map $T$ has a $k$-dimensional invariant subspace, does it admit an $n-k$ invariant subspace? $\lim\limits_{n \to \infty} \frac{a_n}{b_n}=l$ ; Prove $\sum\limits_{k=1}^\infty a_k$ converges iff $\sum\limits_{k=1}^\infty b_k$ converges. Want less brutish proof: if $a+b+c=3abc$ then $\frac1a+\frac1b+\frac1c\geq 3$ Neutral element in $\hom_C(A, B)$