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Let’s determine the convergence of the series ∑ (n=1 to ∞) (n^2 / 2^n). Apply the ratio test. The nth term, a_n = n^2 / 2^n. The (n+1)th term, a_(n+1) = (n+1)^2 / 2^(n+1). Compute the limit: lim (n→∞) |a_(n+1) / a_n| = lim (n→∞) |((n+1)^2 / 2^(n+1)) / (n^2 / 2^n)|. Simplify: lim (n→∞) |(n^2 + 2n + 1) / (2n^2)|. Divide by n^2: lim (n→∞) |(1 + 2/n + 1/n^2) / 2| = 1/2. Since 1/2 < 1, the series converges by the ratio test.
