Question 2
Let $a_1=1$ and $a_{n+1}=a_n+1/\sqrt{a_n}$ for every integer $n\ge1$. Find all real $\alpha$ for which the sequence $u_n=a_n^\alpha/n$ has a finite nonzero limit as $n\to+\infty$.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Importer multipliers: correct 1, incorrect 0, blank 0.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Importer multipliers: correct 1, incorrect 0, blank 0.