Problem 1376 (difficulty: 10/10)

According to Tauber's theorem if \(\displaystyle \displaystyle\lim_{r\to1-0} \sum_{n=0}^\infty a_nr^n=C\) exists and finite and moreover \(\displaystyle n a_n\to0\), then \(\displaystyle \displaystyle \sum_{n=0}^\infty a_n=C\).

(a) Formulate a Tauberian theorem for parametric integrals.

(b) Prove the Tauberian theorem for parametric integrals you formulated.

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