In this paper, we establish the existence of solution for infinite systems of differential equations in the Banach sequence space n ϕ , ℓ p (1 ≤ p < ∞) and c by using Meier-Keeler condensing operators. With the help of examples we illustrate our results in the sequence spaces. Also for validity of the results, we find an approximation of solution by using a suitable method with high accuracy.