An INCREDIBLE Factorial-tastic Infinite Sum of n!²/2n! (Wolfram Alpha can't explain!)

An wonderful infinite sum with an equally spectacular solution, making use of various neat concepts such as Factorials, the Gamma Function, the Beta Function, Fubini's Theorem, a Geometric and Power Series, an interesting technique for Polynomial long division, Integration using Completing the Square, and Trig substitutions!
If you are a fellow math enthusiast, you will be pleased :)
Here's my previous video deriving the solution for the Sum(nxⁿ) mentioned in this tutorial: • An Interesting Geometr...

[@dustydrawer4494]

3 subs and 2 vids is actually insane lol. Same level as maths505 right here. Criminally underrated.

[@dustydrawer4494]

Also, this one has already been covered by dr3123, but I thought it'd be a good one for you to solve on your own time:
sum_(k=1)^(inf)(sink/(k!))

[@Mathority1729]

Appreciate you tremendously for watching, thanks for the support! More videos to come in the near future 🫡

[@Mathority1729]

@@dustydrawer4494 and thanks for the problem, looks interesting, will take a stab at it 👍

[@dustydrawer4494]

​​@@Mathority1729np man! Looking forward to it.

[@dineshmanaria4519]

We can use Taylor series also

[@quasbhop3790]

Incredible video sir!

[@Mathority1729]

Thank you so much!! I truly appreciate you for watching and for the kind comment! 😄

[@fahadibrar379]

Wow

[@Mathority1729]

Thanks for watching! Very much appreciated :)
And more videos to come soon 🫡

[@dineshmanaria4519]

We can solve easily by Taylor series

[@Mathority1729]

Oh that sounds very interesting, could you explain your solution?