EDB — 0DW

Exercises

1. [0DW] Given a series \(∑_ n^∞ a_ n\) tell if the following conditions are necessary and/or sufficient for convergence.

   \begin{eqnarray} ∀\varepsilon {\gt}0~ ∃ m∈{\mathbb {N}}~ ∀ n{\gt}m ~ ∀ k∈{\mathbb {N}}~ ~ \left|∑_{j=n}^{n+k} a_ k\right|{\lt}\varepsilon \\ ∀\varepsilon {\gt}0~ ∀ k∈{\mathbb {N}}~ ∃ m ∈{\mathbb {N}}~ ∀ n{\gt}m ~ \left|∑_{j=n}^{n+k} a_ k\right|{\lt}\varepsilon \\ ∀\varepsilon {\gt}0~ ∃ m∈{\mathbb {N}}~ ∀ n{\gt}m∀ k∈{\mathbb {N}}~ ~ ∑_{j=n}^{n+k} |a_ k| {\lt}\varepsilon \\ ∀\varepsilon {\gt}0~ ∀ k∈{\mathbb {N}}~ ∃ m∈{\mathbb {N}}~ ∀ n{\gt}m ~ ∑_{j=n}^{n+k} |a_ k|{\lt}\varepsilon \end{eqnarray}
   Solution 1

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