r = (2 [Lambda] + [Mu])^2 / (2 ([Mu] + [Lambda])^2);
s  = (1/2 ([Mu]/([Mu] + [Lambda])));
pzero = (1/2) (1 + ([Lambda]/([Mu] + [Lambda]))^2);

pnm = Sum[Binomial[n, k] Binomial[m - 1, n - k - 1] (pzero^k) (r^(n - k)) (s^m),
  {k, 0, n - 1},
  Assumptions -> [Lambda] > 0 && [Mu] > 0 && m > 0 && n > 0 && k >= 0
     && [Lambda] [Element] Integers && [Mu] [Element] Integers
     && m [Element] Integers && n [Element] Integers && k [Element] Integers]
  // FullSimplify;

pn = Sum[pnm, {m, 1, Infinity}] + pzero^n // FullSimplify

Table[pn, {n, 1, 6}] // FullSimplify

*{1,1,1,1,1,1}*

Table[pn, {n, 7, 10}] // FullSimplify // Short