(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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 [tex]\exists[/tex] 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

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