Help needed in understanding the following proof Or please specify the name of this Markov chain

In one of the paper we know the state transition matrix of a Markov chain is given as follows

now to find the steady state probability vector we will have to solve following equation

which can be written in following form

further it is assumed (in the paper)

In the paper it is written that based on above formulation we can obtain the following relations after performing some mathematical manipulations (which I really do not know about and I request you to help me in this)

after that the author says that summing all the term in both sides of above equation and using the fact that ∑∞i=0vi=1\sum_{i=0}^{\infty}v_i=1 we can obtain vkv_k (I do not know how to get there please help me because I get a following form for N=2N=2 p0,1v0+p1,2v1+p2,3v2+p3,4v3+p4,5v4+p5,6v5+….=p2,0v2+p3,1(v2+v3)+p2,0(v3+v4)+p3,1(v4+v5)+p4,2(v5+v6)+…p_{0,1}v_0+p_{1,2}v_1+p_{2,3}v_2+p_{3,4}v_3+p_{4,5}v_4+p_{5,6}v_5+….=p_{2,0}v_2+p_{3,1}(v_2+v_3)+p_{2,0}(v_3+v_4)+p_{3,1}(v_4+v_5)+p_{4,2}(v_5+v_6)+…

and I really do not know how to deal with this infinite sequence on both sides. The author further provides the value of ∑∞k=Nvk\sum_{k=N}^{\infty}v_k as follows

I will be very thankful to you for your help.