For a fixed real parameter β, examine the series R(β)=n=1∑∞n2+1n+β.
(a) Compute n→∞limn2+1n+β in terms of β.
(b) Explain why the result from part (a) by itself neither confirms nor rules out convergence of R(β) for any value of β. Provide a clear argument grounded in the properties of the limit you found.
(c) Compare the general term of R(β) to n1 for large n, and discuss qualitatively what this comparison suggests about the convergence behavior of the series. No detailed proof beyond the discussion is required.