Convergence of Positive Sequences: Limits and Sums from 1 to Infinity

[Askhwhelp]

Suppose a_n > 0 and b_n > 0 for all n in natural number (N). Also, lim a_n/b_n = 0 as n goes to infinity. Then the sum of a_n converges if and only if the sum of b_n converges ...both from 1 to infinity.

My approach is that lim a_n/b_n = 0 means that there exists N in natural number (N) for which |a_n/b_n - 0| < 0 for all n >= N. Then 0 < a_n < 0. The sum of a_n from 1 to infinity is 0. So The sum of a_n from 1 to infinity is convergent.

Is this proof that easy or I miss something?

 

[tiny-tim]

Hi Askhwhelp!

Suppose a_n > 0 and b_n > 0 for all n in natural number (N). Also, lim a_n/b_n = 0 as n goes to infinity. Then the sum of a_n converges if and only if the sum of b_n converges ...both from 1 to infinity.

"if and only if"? … that obviously isn't true

do you mean "if", or do you mean lim a_n/b_n = 1 ?

|a_n/b_n - 0| < 0

sorry, but that doesn't even make sense

check the wording of the question, and start again ​