Calculus of a Single Variable, Chapter 9, 9.4, Section 9.4, Problem 3

Direct comparison test is applicable when suma_n andsumb_n are both positive sequences for all n, such that a_n<=b_n .It follows that:
If sumb_n converges then suma_n converges.
If suma_n diverges then sumb_n diverges.
sum_(n=1)^oo1/(2n-1)
Let b_n=1/(2n-1) and a_n=1/(2n)
1/(2n-1)>1/(2n)>0 for n>=1
As per p series test sum_(n=1)^oo1/n^p is convergent if p>1 and divergent if p<=1
sum_(n=1)^oo1/(2n)=1/2sum_(n=1)^oo1/n
sum_(n=1)^oo1/n is a p-series with p=1, so it diverges.
Since sum_(n=1)^oo1/(2n) diverges ,the series sum_(n=1)^oo1/(2n-1) diverges too by the direct comparison test.

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