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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