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Hi, as you know infinite sum of taylor series may not converge to its original function which means when we increase the degree of series then we may diverge more. Also you know taylor series is widely used for an approximation to vicinity of relevant point for any function. Let's think about a function which is infinitely differentiable so infinitely taylor series exist but it is so complicated function that we can not apply ratio test to find radius of convergence or integral test analytically......In short how we can approximate to any function with a guaranteed convergence except taylor series expansion?