Abel-Dirichlet's Convergence Tests
(Abel) convergent
convergent
bounded
monotone
or
(Dirichlet)
bounded sequence of partial sums
decreasing
convergent
Proof(s):
(Definining partial sum)
(Partial Summation Formula)
For the proofs of both theorems, all we need to do is just to prove that both and are convergent
converge
(Abel)
bounded
monotone
converges
(Monotone convergence theorem)
convergent
converges
converges
bounded
(Convergent sequences are bounded)
bounded
(If we can bound the sum of elements, we certainly can bound the elements themselves)
(Product of absolute values)
(1)
converges
(2)
converges
(Dirichlet)
(Squeeze Theorem)
The second term we just proved upwards