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The first way would be to prove that the function is equal to the series using the Taylor series formula. I assume that's not the issue because it would be trivial. The other way, and they don't often try this before diff eqs, is to find two series that are not the same and both converge on the function of interest. One must have each term larger than the corresponding term in your series, and the other must have each term smaller than the corresponding term. Then as the number of terms approaches infinity, the value of your series is corralled between them. There is no cut and dried method for this. You have to be inspired or lucky.
I'd look at the sum of the first k terms of the Taylor series and try to prove that, in the limit as k goes to infinity, the sum approaches the value of the function. This would probably mean subtracting the (partial) Taylor series from the function and proving that that difference approaches zero. Something like "given epsilon greater than zero, there exists a K for which the sum of the first K terms is within epsilon of the value of f(x) for all x". (Hm. Except that "for all x" is a lot of numbers; maybe that won't work.)
I'm not sure if you're talking about a rigorous mathematical proof or proof that the series approximates the function because the series converges to that function. If your question refers to the latter then you have to try convergence tests that apply to your specific series. ac7zq mentions one such method in his post but there are several other tests depending on the form of the Taylor Series in question. I haven't watched the videos referring to infinite series yet but I'm sure there are several examples of these tests. So the short answer to your question (I think) is that you need to use an appropriate convergence test to show that your series will approximate the function to a certain level of accuracy. Once you prove convergence you set this accuracy by using as many terms of the series as you like. The more terms you use the better your approximation.