Real Analysis Help. Series Convergence

i would love if someone could help me with 2 problems im having trouble with.

1) Let $\displaystyle a_1, a_2, a_3$, ... be a decreasing sequence of positive numbers. Show that $\displaystyle a_1+a_2+a_3+$... converges if and only if $\displaystyle a_1+2a_2+4a_4+8a_8+$... converges. i saw a similar one on here a couple days ago but this is slightly different

2) Show that a power series $\displaystyle \sum\limits_{n = 1}^\infty {c_nx^n }\$ has the same radius of convergence as $\displaystyle \sum\limits_{n = 1}^\infty {c_{n+m}x^n }\$, for any positive integer m

thanks