Non-convergent power series but good approximation?

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Non-convergent power series can still provide good approximations by using only lower-order terms, as they essentially function like polynomials. This phenomenon is recognized in mathematics, particularly in the context of asymptotic expansions, where the relative error can approach zero despite the absolute error diverging. The terms of a divergent series often decrease to a minimum before increasing, which contributes to the effectiveness of the approximation. Understanding specific series, such as Stirling's approximation, can clarify this behavior. Overall, the use of non-convergent series in quantum mechanics demonstrates a legitimate mathematical approach to approximating values.
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Hello,

In my QM class we're using power series which don't converge but apparently still give a good approximation if one only takes the lower-order terms.

Is there any way to understand such a phenomenon? Is it a genuine area of mathematics? Or is it impossible to say something general on this phenomenon?
 
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^ Power series are just polynomials, right? So essentially you're approximating using low-order polynomials. I don't know anything about QM but that'd be my guess as to why what you're doing is reasonable.
 
It might help to look at the math for a particular series to see what is going on. For example http://en.wikipedia.org/wiki/Stirling's_approximation

Usually, the terms of the divergent series reduce in size to a minimum and then increase. If you take the sum up to the smallest term in the series, the relatve error, | approximate value - exact value | / exact value, may converge to 0, even though the absolute error | approximate value - exact value | diverges.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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