Former profile name: i707107

Here is a list of variants of 3 problems that I solved (last edit:5/23/2017):

1. Determine if the sequence converges: $x_0>0$, $x_1>0$, $$ x_{n+2}=\frac{0.2 + x_{n+1}}{0.2 + x_n}. $$ The original problem can be found here: https://math.stackexchange.com/questions/2174888/prove-that-if-x-n2-frac2x-n12x-n-then-x-n-converges/2175064#2175064

2. Determine if the series converges: $$ \sum_{n=1}^{\infty} \frac{|\sin n|\sin(\sqrt 2 n)}n. $$ The original problem can be found here: https://math.stackexchange.com/questions/2202138/determine-whether-sum-n-1-infty-frac-1n-sinnn-converges/2207289#2207289

3. Determine if the series converges: $$ \sum_{p \ \mathrm{prime} } \frac{\sin p}p. $$ The original problem can be found here:https://math.stackexchange.com/questions/2272248/regarding-the-sum-sum-p-textprime-sin-p#comment4674084_2272248