Question

Is the series `2+ 1 + 2/3 + 1/2+ 2/5+ 1/3 +2/7+ ...` convergent or divergent? Explain.

Answer 1

See below.

Explanation:

We have

`S = sum_(k=0)^oo a_k` with `a_k = n_k/d_k`

there exists an internal law to the `a_k` sequence

`n_k/d_k = n_(k-2)/(n_(k-2)+d_(k-2))`

with

`n_1 = 2, d_1 = 1` and
`n_2 = 1, d_2 = 1`

so we have

`0 < a_k < a_(k-2)` and the sequence `a_k` converges.

Concerning the series sum...