Theorem
27
[0DR](Solved on 2022-12-13) Consider the series \(β_{n=1}^β a_ n\) where the terms are positive: \(a_ n{\gt}0\). Define
\[ z_ n = n\left(\frac{a_ n}{a_{n+1}}-1\right) \]
for convenience.
If \(z_ n β€ 1\) eventually in \(n\), then the series does not converge.
If there exists \(L{\gt}1\) such that \( z_ nβ₯ L\) eventually in \(n\), i.e. equivalently if
\[ \liminf _{nββ} z_ n{\gt}1\quad , \]then the series converges.
In addition, fixed \(hβ {\mathbb {Z}}\), we can define
\[ z_ n = (n+h)\left(\frac{a_ n}{a_{n+1}}-1\right) \]
or
\[ z_ n = n\left(\frac{a_{n+h}}{a_{n+h+1}}-1\right) \]
such as
\[ z_ n = n\left(\frac{a_{n-1}}{a_{n}}-1\right) \]
and the criterion applies in the same way.
Solution
1