Proof of the Ratio Test and the Triangle Inequality

[Fiz2007]

Homework Statement

Prove: If the limit inf as k goes to infinity of abs(a_k+1 / a_k) > 1 then the sum from 1 to infinity of a_k diverges

Homework Equations

The Attempt at a Solution

So far I have this:

Suppose lim inf abs(a_k+1/a_k) >1
then, there exists an r such that lim inf abs(a_k+1/a_k) > r > 1
then $$\exists$$ N an element of the natural numbers such that k >= N implies
abs(a_k+1/a_k) > r
that is, for k >= N, abs(a_k+1) > abs(a_k) r
and, abs (a_n+1) > r abs(a_n)
abs (a_n+2) > r abs (a_n+1) > r^2 abs(a_n)
and in general, abs (a_n+k) > r^k abs(a_n)

the series the sum from 1 to infinity of abs(a_n) * r^k diverges to infinity (geometric series with r>1).

Therefore the sum from 1 to infinity of abs(a_n+k) diverges to infinity by the comparison test.

And then I'm stuck... I'm not sure how to go from the absolute value diverging to the series diverging. It clearly does not converge absolutely but what about conditional convergence? Please help!

Thanks

 

[kof9595995]

A necessary condition for convergence of a series is a_n approaches to 0 at infinity.

 

[Fiz2007]

Thanks! that made it much simpler.