Does a Monotonic, Decreasing Series Always Converge to 0?

[talolard]

Hey guys,

Homework Statement

Prove: If for every n a_{n}>0 and \frac{a_{n+1}}{a_{n}}<1 then the series lim_{n->\infty} a_{n}<0

The Attempt at a Solution

We know that a_{n} is lowerly bounded by 0 and upwardly bounded by a_{1}. we also know that it is monotonic and decreasing and so congerges. But how do I show that it converges to 0. What is to stop it from converging to, say, .5?
Thanks
Tal

 

[VeeEight]

When you say series, are you referring to a summation or a sequence?

If a_n < 0 for all n, then all your terms are negative. So apply this fact to a_n+1/a_n < 1

 

[talolard]

Sorry, I made a typo. that was an>0.

 

[talolard]

and I am referring to a sequence, not a summation. Pardon me, english is not my native language.

 

[talolard]

Ahh, I misread the question. it was prove or disprove. I found a counter example.
Thanks anyway.
Tal