Abel-Dirichlet's Convergence Tests
$$ \left\{ a_n \right\}^\infty_{n=1} $$
$$ \left\{ b_n \right\}^\infty_{n=1} $$
(Abel)
$$ \sum_{n=1}^\infty a_n $$
convergent
$$ \left\{ b_n \right\}^\infty_{n=1} $$
bounded
monotone
or
(Dirichlet)
$$ \left| \sum_{n=1}^N a_n \right| \le M $$
bounded sequence of partial sums
$$ b_{n+1} \le b_n $$
decreasing
$$ \lim_{n\rightarrow\infty} b_n = 0 $$
$$ \Rightarrow $$
$$ \sum_{n=1}^\infty a_n b_n $$
convergent
Proof(s):
$$ A_n=\sum_{k=0}^n a_k $$
$$ a_0 = 0 $$
$$ a_n = A_n - A_{n-1} $$
(Definining partial sum)
$$ \sum_{k=1}^n a_kb_k = A_nb_{n+1} - \sum_{k=1}^n A_k\left(b_{k+1} - b_k\right) $$
(Partial Summation Formula)
For proofs of both theorems, all we need to do is just to prove that both
$$ A_nb_{n+1} $$
$$\sum_{k=1}^n A_k\left(b_{k+1} - b_k\right) $$
converge
(Abel)
$$ \left\{ b_n \right\}^\infty_{n=1} $$
bounded
monotone
$$ \Rightarrow $$
converges
(Monotone convergence theorem)
$$ \sum_{n=1}^\infty a_n $$
convergent
$$ \Leftrightarrow $$
$$ \left\{ A_n \right\}^\infty_{n=1} $$
converges
$$ \Rightarrow $$
$$ A_nb_{n+1} $$
converges
$$ \Downarrow $$
$$ \left\{ A_n \right\}^\infty_{n=1} $$
bounded
(Convergent sequences are bounded)
$$ \Downarrow $$
$$ \left\{ a_n \right\}^\infty_{n=1} $$
bounded
(If we can bound the sum of elements, we certainly can bound the elements themselves)
$$ \Leftrightarrow $$
$$ \left| a_n \right| \le M $$
$$ \sum_{k=1}^\infty \left| A_k \left(b_{k+1} - b_k\right) \right| $$
$$ = $$
$$ \sum_{k=1}^\infty \left| A_k \right| \left| \left(b_{k+1} - b_k\right) \right| $$
(Product of absolute values)
$$ \le $$
$$ \sum_{k=1}^\infty M \left| \left(b_{k+1} - b_k\right) \right| $$
$$ = $$
$$ M \sum_{k=1}^\infty \left| \left(b_{k+1} - b_k\right) \right| $$
(1) $ b_{k+1} \ge b_k $
$$ \sum_{k=1}^\infty \left| \left(b_{k+1} - b_k\right) \right| $$
$$ = $$
$$ \sum_{k=1}^\infty \left(b_{k+1} - b_k\right) $$
$$ b_2 - b_1 + b_3 - b_2 + b_4 - b_3 + \cdots $$
$$ = -b_1 $$
$$ \Rightarrow $$
converges
(2) $ b_{k+1} \lt b_k $
$$ \sum_{k=1}^\infty \left| \left(b_{k+1} - b_k\right) \right| $$
$$ = $$
$$ \sum_{k=1}^\infty \left(b_k - b_{k+1}\right) $$
$$ b_1 - b_2 + b_2 - b_3 + b_3 - b_4 + \cdots $$
$$ = b_1 $$
$$ \Rightarrow $$
converges
$$ \blacksquare  $$
(Dirichlet)
$$ \left| \sum_{n=1}^N a_n \right| = \left| A_n \right| \le M $$
$$ \lim_{n\to\infty} b_n = 0 $$
$$ \lim_{n\to\infty} A_nb_{n+1} = 0 $$
(Squeeze Theorem)
The second term we just proved upwards $ \uparrow $
$$ \blacksquare $$
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