EDB — 0F7

Exercises

1. [0F7]Note:This result is attributed to Riemann , see 3.54 in [ 16 ] ..

   Let be given a sequence \((a_ n)_ n\) of real numbers such that \(∑_{n=0}^∞ a_ n\) converges (to a finite value) but \(∑_{n=0}^∞ |a_ n|=∞\); for each \(l,L\) with \(-∞ ≤ l ≤ L ≤ +∞\) there is a permutation \(𝜋:{\mathbb {N}}→{\mathbb {N}}\) such that, defining \(S_ N=∑_{k=0}^ N a_{𝜋(k)}\), we have that

   \[ \limsup _{N→∞} S_ N= L \quad , \quad \liminf _{N→∞} S_ N= l \quad . \]