( arctan x) / ((n^2)+1) n=1 to infinity and n=0 to infinity Analyze the sequence

Question

( arctan x) / ((n^2)+1) n=1 to infinity and n=0 to infinity Analyze the sequence and determine wether it converges or diverges, please show full work to receive 5 stars!! thanks!!

Explanation / Answer

using ratio test we know If for all n, n not equal to 0, then the following rules apply: Let L = lim (n -- >infinity ) | an+1 / an |. If L < 1, then the series an converges. If L > 1, then the series an diverges. hence here an+1/an = [(n+1)^(2) +1] / (n^2)+1) = [ n^2 + 2n +2]/ (n^2 +1) now as n tends to infinity , the limit is 1, hence the test is inconclusive now using integral test integral(( arctan x) / ((n^2)+1)) from n=1 to infinity = (1/4)*pi* arc tanx hence the integral is finite...hence the series converges from n=0 to infinity (1/2)*pi* arc tanx hence the integral is finite...hence the series converges

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