6.4. EXERCISES 161

(a) ∑∞n=1

1√n2+n+1

(b) ∑∞n=1(√

n+1−√

n)

(c) ∑∞n=1

(n!)2

(2n)!

(d) ∑∞n=1

(2n)!(n!)2

(e) ∑∞n=1

12n+2

(f) ∑∞n=1( n

n+1

)n

(g) ∑∞n=1( n

n+1

)n2

2. Determine whether the following series converge and give reasons for your answers.

(a) ∑∞n=1

2n+nn2n

(b) ∑∞n=1

2n+nn22n

(c) ∑∞n=1

n2n+1

(d) ∑∞n=1

n100

1.01n

3. Find the exact values of the following infinite series if they converge.

(a) ∑∞k=3

1k(k−2)

(b) ∑∞k=1

1k(k+1)

(c) ∑∞k=3

1(k+1)(k−2)

(d) ∑∞k=1

(1√k− 1√

k+1

)4. Suppose ∑

∞k=1 ak converges and each ak ≥ 0. Does it follow that ∑

∞k=1 a2

k also con-verges?

5. Find a series which diverges using one test but converges using another if possible.If this is not possible, tell why.

6. If ∑∞n=1 an and ∑

∞n=1 bn both converge and an,bn are nonnegative, can you conclude

the sum, ∑∞n=1 anbn converges?

7. If ∑∞n=1 an converges and an ≥ 0 for all n and bn is bounded, can you conclude

∑∞n=1 anbn converges?

8. Determine the convergence of the series ∑∞n=1(∑

nk=1

1k

)−n/2.

9. Is it possible there could exist a decreasing sequence of positive numbers, {an}such that limn→∞ an = 0 but ∑

∞n=1

(1− an+1

an

)converges? (This seems to be a fairly

difficult problem.)Hint: You might do something like this. Show limx→11−x

− ln(x) =

1−xln(1/x) = 1. Next use a limit comparison test with ∑

∞n=1 ln

(an

an+1

).

10. Suppose ∑an converges conditionally and each an is real. Show it is possible to addthe series in some order such that the result converges to 13. Then show it is possibleto add the series in another order so that the result converges to 7. Thus there is nogeneralization of the commutative law for conditionally convergent infinite series.Hint: To see how to proceed, consider Example 6.2.4.

11. He takes a drug every evening and after 8 hours there is half of it left. Find upper andlower bounds for the amount of drug in his body if he has been taking it for a longtime. Assume each dose consists of 10 mg.