I need help

[dense garnet]

I have completely no idea and even wolfram alpha couldn't calculate this.
Perhaps it might be impossible to calculate this.

[azure dragon]

nothing is impossible

too bad i can’t solve this either

good luck 🥰

[dense garnet]

ty Ill try it again

[distant trellis]

Is that choose function

It's likely the choose function cancels out

Then I would write as partial fractions

This is absurdly hard tbh

Even with the choose function dealt with you have k and n

[quick gull]

Why, it can be simplified. That sum is equal to this.

[dense garnet]

Does choose function mean binomial?
If so,yes

[dense garnet]

Like this?

[dense garnet]

Omg how did you simplify that.
Could you teach me about that?

[quick gull]

There is Gosper's algorithm for sums of similar type. I expressed binomials in terms of factorials, and wolfram alpha gives an answer in terms of gamma function, which can be further simplified to the expression above. As far as I know wolfram uses Gospers algorithm.

[dense garnet]

Wolframalpha could answer it but this is not as brief as your answer.

[dense garnet]

TY Ill learn about Gispers algorithm.

Maybe I ask you some questions about it someday so can you teach me then?

Anyway thank you all very much.🥹

[frail leaf]

sorry to revive this a little but this should be relevant if you want to learn more: https://m.youtube.com/watch?v=0LFg5dvPOoc

the video is a bit rough around the edges but the creator is putting out a follow-up like next week or so going back over the ideas in more detail and clarity