EDB — 0DW

Exercises

1. [0DW] Given a series ${\sum }_n^{\infty }{a}_n$ tell if the following conditions are necessary and/or sufficient for convergence.

   $$\begin{array}{}\text{(1)}& \forall \epsilon >0\exists m\in \mathbb{N}\forall n>m\forall k\in \mathbb{N}\left|{\sum }_{j=n}^{n+k}{a}_k\right|<\epsilon \text{(2)}& \forall \epsilon >0\forall k\in \mathbb{N}\exists m\in \mathbb{N}\forall n>m\left|{\sum }_{j=n}^{n+k}{a}_k\right|<\epsilon \text{(3)}& \forall \epsilon >0\exists m\in \mathbb{N}\forall n>m\forall k\in \mathbb{N}{\sum }_{j=n}^{n+k}|a_k|<\epsilon \text{(4)}& \forall \epsilon >0\forall k\in \mathbb{N}\exists m\in \mathbb{N}\forall n>m{\sum }_{j=n}^{n+k}|a_k|<\epsilon \end{array}$$
   Solution 1

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