Using the ratio test we find that with a(n) = n^2 / 2^n we have
lim(n->infinity)l(a(n+1)/a(n))l =
lim(n->infinity)l(((n + 1)^2 / 2^(n+1)) / (n^2 / 2^n))l =
lim(n->infinity)l((n + 1)/n)^2/ 2)l =
lim(n->infinity)l((1/2)*(1 + (1/n))^2))l = 1/2, so by the ratio test the series
converges absolutely.
Try the integral test
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Verified answer
Using the ratio test we find that with a(n) = n^2 / 2^n we have
lim(n->infinity)l(a(n+1)/a(n))l =
lim(n->infinity)l(((n + 1)^2 / 2^(n+1)) / (n^2 / 2^n))l =
lim(n->infinity)l((n + 1)/n)^2/ 2)l =
lim(n->infinity)l((1/2)*(1 + (1/n))^2))l = 1/2, so by the ratio test the series
converges absolutely.
Try the integral test