Convergence of a sequence implies the convergence of another sequence

Convergence of a sequence implies the convergence of another sequence

Let $a_n>0$, $n¡Ý 0$. Call $k$ ¡°good¡± if $k¡Ý1$ and $a_k>\frac12
a_{k-1}$. Also call 0 good. Show $\sum^{\infty}_{k=0}a_k$ converges if
$\sum_{k \ \ is\ \ good} a_k$ converges. I can only see that the sum of
the good k terms is greater than half of their previous terms' sum, how to
relate the good k terms with every term of this sequence? This is my first
semester in college and first post here, please don't downvote me like
redditors and be gentle! ;-)
-Belen