[Real Analysis] Series of Real Numbers
1. Convergence Tests
Comparion Test
Integral Test
Root and Ratio Test
[Theorem 7.1.8] (Ratio Test)
Let $\sum a_k$ be a series of positive terms and let $R = \overline{\lim\limits_{k \to \infty}} \dfrac{a_{k+1}}{a_k}$, and $r = \lim\limits_{\overline{k \to 0}} \dfrac{a_{k+1}}{a_k}$
(a) If $R < 1$, then $\sum_{k=1}^\infty < \infty$.
(b) If $r > 1$, then $\sum_{k=1}^\infty = \infty$.
(c) If $r \leq 1 \leq R$, then the test is inconclusive.
[Theorem 7.1.9] (Root Test)
Let $\sum a_k$ be a series of nonnegative terms and let $\alpha = \overline{\lim\limits_{k \to \infty}} \sqrt[k]{a_k}$
(a) If $\alpha < 1$, then $\sum_{k=1}^\infty a_k < \infty$
(b) If $\alpha > 1$, then $\sum_{k=1}^\infty a_k = \infty$
(c) If $\alpha = 1$, then the test is inconclusive.