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

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Answer 1 · 2018-02-23T14:40:25

See below.

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...