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.

Then I'm stuck. I don't know how to get from the absolute value diverging to the series diverging. It obvioulsy does not converge absolutely, but what about conditional convergence. Please help!

Thanks

 

[Dick]

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.

Then I'm stuck. I don't know how to get from the absolute value diverging to the series diverging. It obvioulsy does not converge absolutely, but what about conditional convergence. Please help!

Thanks

You've shown the absolute values of the terms in your series are bounded below by a geometric series whose terms go to infinity. The terms in a series have to go to zero if it will have any hope of converging.