Convergent Series: Find Sequence of Positive Reals

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Homework Statement



Find a sequence (an) of positive real numbers such the sum of an from 1 to infinity is convergent but the number of k such that a(k+1)>ak divided by n tends to 1 as n tends to infinity.

Homework Equations





The Attempt at a Solution


I don't have a clue how to find a sequence here. Could anyone please give a hint? Any help is appreciated!
 
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Be creative. Take a nice convergent decreasing series like 1/2^n. Now think about ways of rearranging the terms.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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