Question

physical significance of convergence of series

Answer 1

Now, you have to immediately be able to test whether the summation a.k.a series comes down to a number or goes to infinity so as to be able to solve for xx. If the series is found to be divergent, manipulations by substitution or algebraic modification is done so that the series becomes convergent and the future steps can be solved.

This is why one has to understand convergence and divergence. It is just a tool to check & determine ways of solving the equation to arrive at a solution.