I came across an article (published in IEEE transactions or Wiley publications) which states that moore-penrose generalized inverse can be computed using the following two equations.
Let $A$ be an $m \times n$ matrix. $A^{*}$ is its hermitian conjugate, and $0 < \alpha < c$, where $c$ = max eigenvalue ($A^{*}A$). The convergence of following two equations is assured,
$A^{+} = \alpha \sum_{k=0}^{\infty}(I_{m}-\alpha A^{*}A)^{k} A^{*}$
$A^{+} = \alpha \sum_{k=0}^{\infty}A^{*}(I_{n}-\alpha AA^{*})^{k}$
Through simulations this seems to be true. Can somebody prove that the series is always convergent?
You did not understand the hypothesis. We assume that $\alpha >0$. Note that $c>0$.
Case 1. $rank(A)=n$, that is $A^*A$ is invertible. Then $A^+=(A^*A)^{-1}A^*$ or $A^+=\alpha(I-(I-\alpha A^*A))^{-1}A^*$. We obtain exactly the required series when $\rho(I-\alpha A^*A)<1$, that is $1-\alpha c>-1$, that is $\alpha<2/c$ (your condition is false).
Case 2. $rank(A)=m$, that is $AA^*$ is invertible. Then $A^+=A^*(AA^*)^{-1}$. After, proceed as in Case 1.