Quote by Dick
Sorry! That's wrong. I'm clearly asleep at the wheel. That's convergence for sequences. And this sort of argument only shows that the partial sums are bounded, not that they converge. Do you know the Dirichlet convergence test?

I don't know that one. But the comparison test in my book says the following:
Let [tex]\sum[/tex]a
_{n} be a series where a
_{n} >=0 for all n.
(i) If [tex]\sum[/tex]a
_{n} converges and b
_{n} <= a
_{n} for all n, then [tex]\sum[/tex]b
_{n} converges.
If I let a
_{n} = a
_{n}R
^{n}, this is >=0 for all n. And if I let b
_{n} = a
_{n}(R)
^{n}, then I have b
_{n} <= a
_{n} for all n, so the series converges, right? What's wrong with this statement?