How to find the sum of this series? ∑k=0∞2k(2k+1k)

Kasey Haley

Kasey Haley

Answered

2022-01-22

How to find the sum of this series?
k=02k(2k+1k)

Answer & Explanation

trasahed

trasahed

Expert

2022-01-23Added 14 answers

We have, using the Euler Beta function:
k=0+2k(2k+1k)=k=0+2kΓ(k+1)Γ(k+2)Γ(2k+2)
=k=0+2k(k+1)01xk(1x)kdx
=01dx(12x(1x))2=π2+1
trovabile4p

trovabile4p

Expert

2022-01-24Added 13 answers

Rewrite binomial coefficient with factorials
k=02k(2k+1k)=k=02kk!(k+1)!(2k+1)!
=k=0(k+1)!(2k+1)!
=k=0k!(2k+1)!!+k=0kk!(2k+1)!!
=k=0k!(2k+1)+k=0(k!(2k1)!!(k+1)!(2k+1)!!)
=π2+1

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?