A calculus problem by Kenny Lau

Calculus Level pending

\(a\) and \(b\) are sequences such that:

\(a_0=1\); \(a_n=a_{n-1}\times\dfrac{2n-1}{2n}\)

\(b_n=\dfrac1{2n+1}\)

Find \(\displaystyle\sum_{n=0}^\infty a_nb_n\).

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