EDB — 0DD

Exercises

1. [0DD]Note:Exercise 1 from the written exam 9 April 2011.Let \((a_ n)\) be a sequence of real numbers, with \(a_ n≥ 0\).

   1. Show that if \( ∑_{n=1}^∞ a_ n\) converges then also

      \[ ∑_{n=1}^∞ a_ n^ 2 \quad \hbox{e} \quad ∑_{n=1}^∞ \left(a_ n ∑_{m=n+1}^∞ a_ m\right) \]

      converge.

   2. Assuming moreover that \(∑_{n=1}^∞ a_ n\) is convergent, let’s define

      \[ a=∑_{n=1}^∞ a_ n ~ ~ ,~ ~ b=∑_{n=1}^∞ \left(a_ n ∑_{m=n+1}^∞ a_ m \right)~ ~ ,~ ~ c=∑_{n=1}^∞ a_ n^ 2 \]

      then show that \(a^ 2=2b+c\).