How to Prove Abel's Theorem using Cauchy’s criterion for series and Abel’s formula ?
Abel's Theorem
Let $(a_n)$ and $(b_n)$ be two sequences of real numbers such that
• $(a_n)$ is non-increasing and tends to $0$,
• There exists $M \in R$ such that, for any $n \in N$, $| \sum_{k=1}^{n} b_k| \leq M.$
Then the series $\sum_n a_nb_n$ converges
While Cauchy's criterion for series –
The series $\sum_{n=1}^{\infty} a_n $ converges if and only if for every $\epsilon > 0 $ there is a positive integer N such that if m > n > N then $| \sum_{j=n}^{m} a_n | < \epsilon $
How can I prove the above using these both.
Best Answer
You also need $a_n>0$ for this to work, so I'll assume it from now on.
Let $$B_n=\sum_{k=0}^n b_k$$
Then $$\begin{align} \sum_{k=n}^{m} a_k b_k & = \sum_{k=n}^m a_k (B_k-B_{k-1}) \\ & = \sum_{k=n}^m a_k B_k - \sum_{k=n}^m a_k B_{k-1} \\ & = \sum_{k=n}^m a_k B_k - \sum_{k=n-1}^{m-1} a_{k+1} B_k \\ & = a_m B_m - a_nB_{n-1} +\sum_{k=n}^{m-1}(a_k - a_{k+1})B_k \end{align}$$
So $$\begin{align} \left|\sum_{k=n}^{m} a_k b_k\right| & \leq a_m |B_m| + a_n |B_{n-1}| + \sum_{k=n}^{m-1} (a_k-a_{k+1})|B_k| \\ & \leq M(a_m + a_n + a_n - a_m) \\ & \leq 2M a_n \end{align}$$
Since $a_n$ tends to $0$, then our series satisfies the Cauchy Criterion.