So I was working on a problem where I am provided with a PMF $p_X(k)= c/3^k$ for $k=1,2,3....$ I was able to calculate $c$ using the basic property of PMF and it came to be 2. I am not able to solve the next part which states that "Find $P(X\ge k)$ for all $k=1,2,3......$.
Any suggestions?
P.S :Here is the actual question:
Let X be a discrete random variable with probability mass function $p_X(k) = c/3^k$ for k = 1, 2, ... for some $c > 0$. Find $c$. Find $P(X\ge k)$ for all $k = 1, 2,3....$
Consider, $$S=\frac{2}{3}+\cdots+\frac{2}{3^{k-2}}+\frac{2}{3^{k-1}}$$
multiply $S$ by $\frac{1}{3}.$ Thus,
$$\frac{1}{3}S=\frac{2}{3^{2}}+\cdots+\frac{2}{3^{k-1}}+\frac{2}{3^{k}}.$$
Subtract $S$ of $\frac{1}{3}S,$
$$\frac{2}{3}S=\frac{2}{3}-\frac{2}{3^{k}}.$$ Thus,
$$S=1-\frac{1}{3^{k-1}}.$$ Now,
if $k=1, P(X\geq k)=1,$ and if $k>1,$
\begin{eqnarray} P(X\geq k)&=&1-P(X< k)\\ &=&1-P(X\leq k-1)\\ &=&1-\displaystyle \sum_{n=1}^{k-1}\frac{2}{3^{n}}\\ &=&1-(1-\frac{1}{3^{k-1}})\\ &=&\frac{1}{3^{k-1}} \end{eqnarray}
