Given {$${{a_n}}$$} is decrease and positive, and
series $$\sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^n}{a_n}} $$
is divergent.
set $${u_n} = (\frac{1}
{{1 + {a_1}}})(\frac{1}
{{1 + {a_2}}})...(\frac{1}
{{1 + {a_n}}})$$
Question: the series $$\sum\limits_{n = 1}^\infty {{u_n}} $$
is divergent or convergent, and why.

Question

Given {$${{a_n}}$$} is decrease and positive, and
series $$\sum\limits_{n = 1}^\infty  {{{\left( { - 1} \right)}^n}{a_n}} $$
is divergent. 
set $${u_n} = (\frac{1}
{{1 + {a_1}}})(\frac{1}
{{1 + {a_2}}})...(\frac{1}
{{1 + {a_n}}})$$
Question: the series $$\sum\limits_{n = 1}^\infty  {{u_n}} $$
is divergent or convergent, and why.

anonymous · Answer

Please somebody help... I am desperate..