# probability and series

$X$ is a discrete random variable taking on the values $X=1,3,3^2,3^3,\ldots,3^m$ and $f(x)=P(X=x)=c / x$ for a constant $c$. Find $c$.

Solution: Since $P(X)=1$, we know that $cx=1$, so $c=x$. To find $x$, we have $x=\sum_{k=0}^m 3^m$. Since this series summation diverges to infinity, $c=∞$. This is a fascinating problem, however, something doesn't seem right ... in other words, how can $x=∞$? Is the first statement $c=x$ incorrect? Any help is greatly appreciated

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$P(X)$ is nonsense. $P(X=x)$ has a meaning - it's the probability with which $X$ takes on the value $x$. You are told that the values that $X$ takes on are $1,3,3^2,\dots,3^m$, and that if $x$ is any one of these numbers then the probability with which $X$ takes on that value is $c/x$.
Now what you know about probabilities is that if you add up all the numbers $P(X=x)$ over all the possible values of $x$ you get 1. Now that I've cleared up a few things (I hope), can you take it from there?
@John Note quite: $c/x$ is not equal to $(c/1)+(c/3)+\cdots+(c/3^m)$; but the latter quantity is equal to $1$. – David Mitra Apr 30 '12 at 4:53