So you know . What can you say about ? Does it converge? Where to? You also know that . So then what can you say about the sum that you found?
The following problem is in a section on convergence tests for series (comparison, ratio, root, integral, p-series, etc.) I've been stuck on it for about a day, so any help would be much appreciated.
Suppose that is a convergent series with positive terms. Let be the sequence of partial sums for . Let and let . Suppose there is a number for which if and . Show that .
So far all I've been able to do is show that which means I need to show that , but I'm not sure where to take it from here, or if this is even a good approach.