Three Series for $1/\pi^2$ and $1/\pi^3$ (Open)

Summary: The proposer identifies three series for reciprocal powers of $\pi$ that are found in fairly recent papers of J. Guillera. Two of the series (each for $1/\pi^2$) have been proved using an ingenious application of the WZ method, while the third series (for $1/\pi^3$) is unproved.
Problem 1. Find non-WZ proofs of the first two series.
Problem 2. Prove the third series.

Classification: Primary, Classical Analysis; Secondary, Sequences and Series

Jonathan Borwein
Faculty of Computer Science
Dalhousie University
Halifax, NS B3H 1W5
Email: [email protected]
Phone: 902-494-2501
Fax: 902-494-7710

Currently visiting the University of Newcastle

Jonathan Borwein
Visiting Professor Laureate
School of Mathematical and Physical Sciences
University Drive
University of Newcastle
Callaghan NSW 2308, Australia
Email: [email protected]
Phone: (02) 4921 5535
Fax: (02) 4921 6898

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