Could someone please help me understand Euler's series acceleration?

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I am trying to accelerate a Taylor series with an alternating sign and found that Euler's acceleration is something I should look into.

I found this document,

https://kconrad.math.uconn.edu/blurbs/analysis/series_acceleration.pdf

which in section 3 pretty clearly explains the math behind it until point 3.20, after which I find it really hard to understand what is going on.

Could someone help me understand it.

Thank you in advance.