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Calculate the ratio of the (n+1)'th term to the n'th term and then let n tend to infinity. You will find in each case that the ratio tends to 1, so that there are an infinite number of same size terms being added, leading to an infinite sum.

For example, the ratio of the (n+1)'th to the n'th term for your first sum is: (n+2)(n^2+2n+3)/[(n+1)(n^2+4n+6)] which tends to 1 as n tends to infinity.

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Alan Apr 25, 2017