Let $x=\frac{22}{7} \in \mathbb{Q}$.
(a) Find $\mathrm{ord}_p(x)$ for all primes $p$.
$\mathrm{ord}_2(x)=1,\ \mathrm{ord}_{11}(x)=1,\ \mathrm{ord}_7(x)=-1$ and $\mathrm{ord}_p(x)=0$ for all other primes.
(b) Determine $|x|_p=p^{-\mathrm{ord}_p(x)}$ for all primes $p$.
$|x|_2=\frac{1}{2}$, $|x|_{11}=\frac{1}{11}$, $|x|_7=7$ and $|x|_p=1$ for all other primes.
(c) Hence determine for which primes $p$ the geometric series $1+x+x^2+\ldots$ converges in $\mathbb{Q}_p$ and what is the sum when it converges.
I know a series with terms $\in \mathbb{Q}_p$ converges if and only if the terms tend to zero.
Ok, you know enough. You need to know, when $|x^n|_p\to 0$. But
1. $|x^n|_p=|x|_p^n$
2. for $a\in \mathbb R_{>0}$ occurs $a^n\to 0\Leftrightarrow a<1$.
As a result, we have that it happens only for $p\in\{2, 11\}$.