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A question from Rudin (Principles) Chapter 3:

Let $a_n\geq0$ and $\sum a_n$ diverges. What can be said about convergence/divergence of $\sum\frac{a_n}{1+na_n}$?

This one is being recalcitrant. Given that $x>y$ implies $\frac{x}{1+nx}>\frac{y}{1+ny}$ and when $a_n=1/n\log n$ the sum in question diverges, it seems plausible that in general the sum will always diverge, but I can't get a proof out. If it does diverge, it does so pretty slowly as $$\frac{a_n}{1+na_n}=\frac{1}{n}-\frac{1}{n+n^2a_n}\leq\frac{1}{n}.$$

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    I'd try some example divergent sequences first. $a_n=1$ the new series does not converge. Same for $a_n=\frac{1}{n}$.2012-04-11

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