We're going to mess around with some divergent series in this video, namely the Grandi Series, 1-1+1-1+... and the related series 1-2+3-4+... and I'll hopefully convince you that the sums that are associated with these divergent series make some sense when seen through the lens of calculus, using nothing more than a Taylor series.
The method is called analytic continuation, but hopefully I can give you an overview of the concept without worrying about hardcore complex analysis, since the topic of analytic continuation is mentioned in passing in the famous (notorious) Numberphile video and also covered in Mathologer's recent video. I believe it would be beneficial to provide at least one example to illustrate the concept and give yet another example of the power of calculus.
Here's a link to an article which is a nice primer on the topic:
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And here's a link to Terry Tao's (mathematician extraordinaire) blog post on the topic for the adventurous:
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