I have the following transition matrix for my Markov Chain:
$$ P= \begin{pmatrix} 1/2 & 1/2 & 0&0&0& \cdots \\ 2/3 & 0 & 1/3&0&0&\cdots \\ 3/4 & 0& 0&1/4&0&\cdots \\ 4/5 & 0&0&0&1/5&\cdots\\ 5/6&0&0&0&0&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots \end{pmatrix} $$
defined by
$$ P_{ij} = \begin{cases} \frac{i}{i+1}, & \text{if j=1} \\ \frac{1}{i+1}, & \text{j=i+1}\\ 0, & \text{otherwise.} \end{cases} $$
I've tried finding a pattern for just a 5x5 version of this but I can't seem to find anything unique. In fact, all of these zeroes seem to take on values at higher powers of the matrix.
The only pattern I see is that each element in the 1st column becomes the sum of 1/2 the element in the column plus 1/i+1. They seem to approach .582 for all elements... But how would I show this in the infinite case? I can't envision how I would multiply by the stationary distribution $\pi$.
Can anyone give me some insight on where I should go from here? Thank you greatly in advance!
EDIT: Forgot the actual question. Ugh I'm tired :/
(a) Does the chain have a stationary distribution? If yes, exhibit the distribution. If no, explain why.
(b) Classify the states of the chain.