*This series is divergent; therefore, we may be able to do something with it.* -- Oliver Heaviside


[Edit (1/14/21) from the answer by Count Iblis to a recent MO-Q on math vids: An enthusiastic intro is that to the set of lectures by Carl Bender "[Perturbation and Asymptotic Series](https://www.youtube.com/playlist?list=PL43B1963F261E6E47)." ]


Other than the usual references given in Wikipedia and Mathworld, which resources have you found helpful as intros to the topic and for advanced exploration?


I'll prime the pump with


1. "[Divergent series:taming the tails](https://michaelberryphysics.files.wordpress.com/2013/06/berry482.pdf)" by M. V. Berry and C. J. Howls (cf. also refs in this [MO-Q](https://mathoverflow.net/questions/227642/investigation-of-sum-limits-k-infty-infty-fracxkn-gammakn1/227660#227660))
2. Sporadic examples in Heaviside's publications, see [Heaviside's Operational Calculus](https://www.google.com/amp/s/deadreckonings.com/2007/12/07/heavisides-operator-calculus/amp/), a post by Ron Doerfler.
3. [A Singular Mathematical Promenade](https://arxiv.org/abs/1612.06373) by Etienne Guys
4. [Sum Divergent Series](https://cornellmath.wordpress.com/2007/07/28/sum-divergent-series-i/) by the user mnoonan, a series of posts at The Everything Seminar
5. "[Euler's constant:
Euler's work and modern developments](https://arxiv.org/abs/1303.1856)" by Jeffrey Lagarias
6. "[Uniform asymptotic methods for integrals](https://www.sciencedirect.com/science/article/pii/S001935771300061X)" by Nico Temme
7. "[On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)](http://philsci-archive.pitt.edu/2629/)" by
Robert W. Batterman


For one example of the importance of such series, see the relation between the Harer-Zagier formula and the asymptotic expansion of the digamma function in Chapter 5 "The Euler characteristic of the moduli space of curves" of the course notes "[Mathematical ideas and notions of quantum field theory](https://math.berkeley.edu/%7Elinlin/2017Spring_290/ETI.pdf)" by Etingof.



Resources for divergent / asymptotic series
As far as on-line-available things go, I've attempted to modernize some arguments and give examples of asymptotics of integrals (both Watson's Lemma and easy Laplace/saddle-point examples), as well as asymptotics for ordinary differential equations, both regular and certain irregular singular points. On-line, as well as a chapter in my Cambridge Univ Press book of 2018 ([http://www.math.umn.edu/~garrett/m/v/current\_version.pdf](http://www.math.umn.edu/%7Egarrett/m/v/current_version.pdf)). For earlier, separate treatments, see [http://www.math.umn.edu/~garrett/m/mfms/notes\_2019-20/05e\_asymptotics\_of\_integrals.pdf](http://www.math.umn.edu/%7Egarrett/m/mfms/notes_2019-20/05e_asymptotics_of_integrals.pdf), [http://www.math.umn.edu/~garrett/m/mfms/notes\_2013-14/11b\_reg\_sing\_pt.pdf](http://www.math.umn.edu/%7Egarrett/m/mfms/notes_2013-14/11b_reg_sing_pt.pdf), and [http://www.math.umn.edu/~garrett/m/mfms/notes\_2013-14/11c\_irreg\_sing\_pt.pdf](http://www.math.umn.edu/%7Egarrett/m/mfms/notes_2013-14/11c_irreg_sing_pt.pdf). Those notes (and the book, on-line or not) have substantial bibliographic/historical references.


