- #1
talolard
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Hey guys,
Prove: If for every n [tex] a_{n}>0 [/tex] and [tex] \frac{a_{n+1}}{a_{n}}<1 [/tex] then the series [tex]lim_{n->\infty} a_{n}<0 [/tex]
We know that [tex] a_{n} [/tex] is lowerly bounded by 0 and upwardly bounded by [tex] a_{1} [/tex]. 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
Homework Statement
Prove: If for every n [tex] a_{n}>0 [/tex] and [tex] \frac{a_{n+1}}{a_{n}}<1 [/tex] then the series [tex]lim_{n->\infty} a_{n}<0 [/tex]
The Attempt at a Solution
We know that [tex] a_{n} [/tex] is lowerly bounded by 0 and upwardly bounded by [tex] a_{1} [/tex]. 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
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