64 CHAPTER 4. FUNCTIONS AND SEQUENCES

4.7 Exercises1. Find limn→∞

n3n+4 .

2. Find limn→∞3n4+7n+1000

n4+1 .

3. Find limn→∞2n+7(5n)4n+2(5n) .

4. Find limn→∞

√(n2 +6n)−n. Hint: Multiply and divide by

√(n2 +6n)+n.

5. Find limn→∞ ∑nk=1

110k .

6. Suppose {xn + iyn} is a sequence of complex numbers which converges to the com-plex number x+ iy. Show this happens if and only if xn→ x and yn→ y.

7. For |r|< 1, find limn→∞ ∑nk=0 rk. Hint: First show ∑

nk=0 rk = rn+1

r−1 −1

r−1 . Then recallTheorem 4.4.11.

8. Using the binomial theorem prove that for all n ∈ N,(1+

1n

)n

≤(

1+1

n+1

)n+1

.

Hint: Show first that(n

k

)= n·(n−1)···(n−k+1)

k! . By the binomial theorem,

(1+

1n

)n

=n

∑k=0

(nk

)(1n

)k

=n

∑k=0

k factors︷ ︸︸ ︷n · (n−1) · · ·(n− k+1)

k!nk .

Now consider the term n·(n−1)···(n−k+1)k!nk and note that a similar term occurs in the

binomial expansion for(1+ 1

n+1

)n+1except you replace n with n+1 whereever this

occurs. Argue the term got bigger and then note that in the binomial expansion for(1+ 1

n+1

)n+1, there are more terms.

9. Prove by induction that for all k ≥ 4, 2k ≤ k!

10. Use the Problems 21 and 8 to verify for all n ∈ N,(1+ 1

n

)n ≤ 3.

11. Prove limn→∞

(1+ 1

n

)nexists and equals a number less than 3.

12. Using Problem 10, prove nn+1 ≥ (n+1)n for all integers, n≥ 3.

13. Find limn→∞ nsinn if it exists. If it does not exist, explain why it does not.

14. Recall the axiom of completeness states that a set which is bounded above has a leastupper bound and a set which is bounded below has a greatest lower bound. Show thata monotone decreasing sequence which is bounded below converges to its greatestlower bound. Hint: Let a denote the greatest lower bound and recall that because ofthis, it follows that for all ε > 0 there exist points of {an} in [a,a+ ε] .