# convergent and divergent series indeterminate by ratio and root test

(i) Give an example of a divergent series $\sum\limits_{n=1}^{\infty} a_n$ of positive numbers $a_n$ such that $$\lim\limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim\limits_{n \to \infty} a^{1/n} = 1$$

Answer: $\sum\limits_{n=1}^\infty 1^n$

(i) Give an example of a convergent series $\sum\limits_{n=1}^{\infty} a_n$ of positive numbers $a_n$ such that $$\lim\limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim\limits_{n \to \infty} a^{1/n} = 1$$

Answer: $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

• Yes except there is an extra ''='' sign after the $\sum$ sign. – gmath Nov 4 '14 at 18:16
• A simpler way to write the $n^{th}$ term of the first series: $1$. – Jonas Meyer Nov 4 '14 at 18:59