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

 

[CellCoree]

Thank you for your question. The phenomenon you are describing is known as asymptotic series or divergent series. These are power series that do not converge, but can still provide a good approximation by taking only a finite number of terms. This is a common occurrence in many areas of mathematics and physics.

The reason why these series can still provide a good approximation is due to the fact that they have a specific structure. In particular, the terms in the series decrease in magnitude as the index increases. This means that the higher order terms have less and less impact on the overall sum, allowing us to truncate the series at a finite number of terms without significantly affecting the accuracy of the approximation.

However, it is important to note that this does not always hold true and the accuracy of the approximation can vary depending on the specific series and the function it is approximating. Therefore, it is always important to carefully consider the convergence and accuracy of a series before using it as an approximation.

As for whether this is a genuine area of mathematics, the answer is yes. Asymptotic series have been studied extensively and have many applications in various fields such as physics, engineering, and economics. However, it is a complex and ongoing area of research and there is still much to be understood about these types of series.

In conclusion, while non-convergent power series may seem counterintuitive, they can still provide useful approximations in certain cases. It is a fascinating area of mathematics and one that continues to be explored by scientists and mathematicians alike.