Use the limit comparison test to determine if sum, from, n, equals, 1, to, infinity, of, start fraction, 3, n, start superscript, start fraction, 5, divided by, 2, end fraction, end superscript, plus, 6, divided by, 2, n, cubed, minus, 2, n, squared, plus, 5, end fractionn=1∑∞​2n3−2n2+53n25​+6​ converges or diverges, and justify your answer.

Answer

Attempt 1 out of 10

Apply the comparison test with the series sum, from, n, equals, 1, to, infinity, of, start fraction, 1, divided by, n, to the power p , end fractionn=1∑∞​np1​ where p, equalsp= . If a, start subscript, n, end subscript, equals, start fraction, 3, n, start superscript, start fraction, 5, divided by, 2, end fraction, end superscript, plus, 6, divided by, 2, n, cubed, minus, 2, n, squared, plus, 5, end fractionan​=2n3−2n2+53n25​+6​ and b, start subscript, n, end subscript, equals, start fraction, 1, divided by, n, to the power p , end fractionbn​=np1​, then limit, start subscript, n, right arrow, infinity, end subscript, start fraction, a, start subscript, n, end subscript, divided by, b, start subscript, n, end subscript, end fraction, equalsn→∞lim​bn​an​​= . Since a, start subscript, n, end subscript, comma, b, start subscript, n, end subscript, is greater than, 0an​,bn​>0 and the limit is a finite and positive (non-zero) number, the limit comparison test applies. sum, from, n, equals, 1, to, infinity, of, start fraction, 1, divided by, n, to the power p , end fractionn=1∑∞​np1​  convergesdiverges since a p-series will ______ if and only if   p  <  1 p ≤ 1 p > 1 p ≥ 1. Therefore,sum, from, n, equals, 1, to, infinity, of, start fraction, 3, n, start superscript, start fraction, 5, divided by, 2, end fraction, end superscript, plus, 6, divided by, 2, n, cubed, minus, 2, n, squared, plus, 5, end fractionn=1∑∞​2n3−2n2+53n25​+6​  convergesdiverges.