Answers and comments for Math 242 Homework 9.


#1. Yes, it's a decreasing function.

It's acceptable to say: We know a rule that says
ln x grows slower than x^a.

For a more precise answer, notice that f'(x) is

( 1 - ln x )/x^2, which is negative if x > e.

So f(x) is decreasing if x > e.


#2. Converges.

Alternating series test. We know from earlier
that (ln n)/n decreases, and it approaches 0.


#3. Converges.

Can use basic comparison: a_n < n/(n4^n) = 1/4^n
or can use ratio test.


#4. Converges.

Alternating series test. Note that when n
approaches infinity, Pi/n is positive and
decreases to zero. It follows that sin(Pi/n)
is positive and decreases to zero.


#5. Converges.

Can use basic comparison. Notice that

-1 < sin n < 1

0 < 1 + sin n < 2

0/10^n < (1 + sin n)/10^n < 2/10^n

The sum of 2/10^n is a geometric series
with r = 1/10, so it converges.


#6. Diverges.

Can use pre-test. Notice that
sqrt(n)/(1+2sqrt(n)) approaches 1/2.
Therefore, the terms of this series
do NOT approach zero.


#7. Diverges.

Can use ratio test. (Should get L = 1.1)


#8. Converges conditionally.

First, the given series converges "as is",
because of the alternating series test.
(Roughly speaking, the positive part of the
nth term is "like" n/sqrt(n^4) = n/n^2 = 1/n.)

What if we "make" all the terms positive?
We expect that series to diverge.
We can prove this e.g. by using limit comparison,
with b_n = 1/n.


#9. Converges absolutely.

Can use ratio test. Should get L=0.