Non-convergent power series but good approximation?

[nonequilibrium]

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?

 

[Dr. Seafood]

^ 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.

 

[AlephZero]

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.

 

[awkward]

Such an approximation is called an "asymptotic expansion".

http://en.wikipedia.org/wiki/Asymptotic_expansion