homework help...calculus problems?

Question

Determine whether the sequence converges or diverges. If it converges find the limit.

1) (3 + 5n^2)/(n + n^2)

2) an= n/(1+ radical n)

3) an= (-1)^(n-1)n/(n^2 + 1)

4) {n^2e^-n}

Please either explain how to do it or show me the steps so i know..thanks! :)

Answer 1

For sequences like these, it is usually best to find the limit as n approaches infinity.

1) divide everything by n^2, then find limit as n-->infinity. It should be 5, meaning the sequence converges.

2) Divide everything by sqrt(n), then find lim n-->infinity.

3) The numerator will be either +1 or -1, depending upon n, but the denominator keeps getting larger.

4) Rewrite this as n^2/e^n. Sure, 1,000,000^2 is big, but e^1,000,000 is even bigger. This sequence converges to 0 very quickly.