From Thomas, 9th edition, Page 700, 2-40 evens. Note that other correct reasons exist for these answers.

2) lim_n®¥ a_even = lim_n®¥ [2/(Ön)] = 0 and lim_n®¥ a_odd = 0 so lim_n®¥ a_n = 0

4) lim_n®¥ a_n = 1 + 0 = 1

6) lim_n®¥ a_n = lim_n®¥ 0 = 0

8) lim_n®¥ a_n = 0 by l'Hôpital.

10) lim_n®¥ a_n = 0 by l'Hôpital.

12) lim_n®¥ a_n = 1/e by logarithmic differentiation.

14) lim_n®¥ a_n = 1 by logarithmic differentiation.

16) lim_n®¥ a_n = 1 by logarithmic differentiation.

18) lim_n®¥ a_n = 0 as 5ⁿ < n! for n ³ 12 thus 0 £ | lim_n®¥ a_n| £ lim_n®¥ (4/5)ⁿ = 0

20) åa_n = -2 å(1/n - 1/(n+1) = -2(1/2 + 0) = -1

22) åa_n = -2 å(1/(4n-3)- 1/(4n+1) = -2(1/9 + 0) = -2/9

24) åa_n = -3/5 by geometric series.

26) åa_n divergent by limit comparison test with 1/n.

28) åa_n absolutely convergent by the integral test.

30) åa_n absolutely convergent by the integral test.

32) åa_n divergent by direct comparison test with 1/n.

34) åa_n is not absolutely convergent by limit comparison test with 1/n, but is conditionally convergent by the alternating series test.

36) åa_n divergent since lim_n®¥ a_n = 1/2

38) åa_n is absolutely convergent by the nth root test.

40) åa_n is absolutely convergent by the limit comparison test with 1/n²