Prove the following Comparison test:
Let ∑ k = 0 ∞ a k {\displaystyle {}\sum _{k=0}^{\infty }a_{k}} and ∑ k = 0 ∞ b k {\displaystyle {}\sum _{k=0}^{\infty }b_{k}} be two series of non-negative real numbers. The series ∑ k = 0 ∞ b k {\displaystyle {}\sum _{k=0}^{\infty }b_{k}} is divergent and moreover we have a k ≥ b k {\displaystyle {}a_{k}\geq b_{k}} for all k ∈ N {\displaystyle {}k\in \mathbb {N} } .
and 
be two series of non-negative real numbers.
The series is divergent and moreover we have
for all
.
