Which of the following facts are true of a sequence satisfying $\lim a_n^{\frac{1}{n}}=1$?

sequences-and-series convergence-divergence

Solution 1:

First of all, way to go for your efforts. As far as I can see, your answers to the first three questions are correct. To refute the last one consider the sequence

$$\{a_n\}=\{1,1,2,1,3,1,4,1,5,1,6,1,....\}=\begin{cases}k,&n=2k-1\\{}\\1,&\text{otherwise}\end{cases}$$

Observe that

$$\left\{\frac{a_{n+1}}{a_n}\right\}=\left\{1,2,\frac12,3,\frac13,4,...\right\}$$

Solution 2:

Your answer to 2) is wrong. $R=1$ implies the power series converges absolutely for $|x|<1,$ but not uniformly in that range. Example: $\sum x^n.$ However, the power series will converge uniformly in $[-a,a]$ for all $a\in [0,1).$

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