(n+2)/(n+1) converges as n tends to infinity.

It gets arbitrarily close to 1 without ever getting there. If you needed to do a formal proof, you would show that for any value epsilon>0, there exists N such that for all n>N, (n+2)/(n+1) differs from 1 by less than epsilon.

The integral of (n+2)/(n+1) diverges becauses it's adding up an infinite series of numbers all of which are greater than 1.

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