5.3. EXERCISES 89

Example 5.2.12 Determine the convergence of ∑∞k=1

1√n2+100n

.

Use the limit comparison test. limn→∞

( 1n )(1√

n2+100n

) = 1 and so this series diverges along

with ∑∞k=1

1k .

Sometimes it is good to be able to say a series does not converge. The nth term testgives such a condition which is sufficient for this. It is really a corollary of Theorem 5.1.7.

Theorem 5.2.13 If ∑∞n=m an converges, then limn→∞ an = 0.

Proof: Apply Theorem 5.1.7 to conclude that limn→∞ an = limn→∞ ∑nk=n ak = 0.

It is very important to observe that this theorem goes only in one direction. That is,you cannot conclude the series converges if limn→∞ an = 0. If this happens, you don’tknow anything from this information. Recall limn→∞ n−1 = 0 but ∑

∞n=1 n−1 diverges. The

following picture is descriptive of the situation.

∑an converges

liman = 0

an = n−1

5.3 Exercises1. Determine whether the following series converge and give reasons for your answers.

(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) ∑Nk=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?