Can you give examples of 2 series that satisfy the following conditions?

Question

Prove that the ratio test is inconclusive when the lim as n->∞ of |a_{n+1} / a_n|=1.

∞ of |1/(n+1)÷1/n| = the lim as n->∞ of |n/(n+1)| = 1.

Answer 1

A convergent series with this property is the ∑(-1)^n/n from n = 1 to ∞,

since the lim as n -> ∞ of | (-1)^(n+1)/(n+1) ÷ (-1)^(n)/n | =

the lim as n -> ∞ of | (-1)^(n+1) * n / [(n+1) * (-1)^n] | =

the lim as n -> ∞ of | (-1)^(1) * n / (n+1) | = 1;

and a divergent series with this property is the ∑(1/n) from n = 1 to ∞,

since the lim as n->∞ of |1/(n+1)÷1/n| = the lim as n->∞ of |n/(n+1)| = 1.