> It seems then no coincidence that in physics and math we often consider partial sums as approximations, and concentrate on particular terms seperately. In physics for example, people think of "order \alpha^2 terms" or "higher order corrections" or physical quantities which are often taylor series cut off at some order, of course, assuming the quantity.
There are very good reasons for this. The whole observation is basically the same phenomenon as someone observing, "you know, 34,825,119,276 and 35,174,884,395 are basically the same number for my purposes; I'll just call it 3e10 or, if I'm being really fancy, 3.5e10".
In these applications, the series variable is a very small number. The higher exponents given to it in later terms of a taylor series cause those terms to be very small compared to the early terms. That's why we take the early terms as an approximation to the whole thing -- we have chosen our representation so that that will be true.
(This is the entire reason for Taylor series in the first place -- a Taylor series is a Maclaurin series adjusted so that the variable can be small for purposes of the series, no matter what its absolute value might be.)
Yes, I think the parent realises that. The point I think is that this stuff would be much harder to realise if we used verbose and prosaic descriptions instead of the visually suggestive modern notation (implicit multiplication etc)
As I read the parent comment, it suggests that the reason we think of the first few terms of a Taylor series as approximating the whole thing is that the notation suggests that the series is composed of a sequence of discrete terms. We know that this is wrong; Taylor series were developed so that the first few terms would approximate the whole.
There are very good reasons for this. The whole observation is basically the same phenomenon as someone observing, "you know, 34,825,119,276 and 35,174,884,395 are basically the same number for my purposes; I'll just call it 3e10 or, if I'm being really fancy, 3.5e10".
In these applications, the series variable is a very small number. The higher exponents given to it in later terms of a taylor series cause those terms to be very small compared to the early terms. That's why we take the early terms as an approximation to the whole thing -- we have chosen our representation so that that will be true.
(This is the entire reason for Taylor series in the first place -- a Taylor series is a Maclaurin series adjusted so that the variable can be small for purposes of the series, no matter what its absolute value might be.)