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: Prove that if $x_{n+2}=\frac{2+x_{n+1}}{2+x_n},$ then $x_n$ converges

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: Determine whether $\sum_{n=1}^\infty \frac {(-1)^n|\sin(n)|}{n}$ converges

3. Determine if the series converges: $$ \sum_{p \ \mathrm{prime} } \frac{\sin p}p. $$ The original problem can be found here:Regarding the sum $\sum_{p \ \text{prime}} \sin p$