Use the comparison test to determine whether the series converges or diverges.
Answer
This series looks like , so we guess it diverges. Since making the denominator smaller makes the fraction bigger,
(for n > 5, anyway). Since the harmonic series diverges and the series
has bigger terms, this series has to diverge too. The comparison test tells us so.
Example 2
Use the comparison test to determine whether the series converges or diverges.
Answer
For n ≥ 2 we know that
The series
converges because it's geometric with . Since the series
has smaller terms, the comparison test says this series must converge also.
Example 3
Use the comparison test to determine whether the series converges or diverges.
Answer
Making the denominator smaller makes the fraction bigger, so
The series
is geometric with ratio , so it converges. The comparison test says the series
which has smaller terms, has to converge too.
Example 4
Use the comparison test to determine whether the series converges or diverges.
Hint
for n ≥ 1
Answer
This one's a little sneaky. Do the terms look like or like ? Either way, we guess the series diverges. To show this we need to find a divergent series with smaller terms. If we make the denominator bigger we make the fraction smaller, so replace with n:
Since multiplication by a constant doesn't affect whether the series diverges,
diverges. The comparison test says that
which has bigger terms, also diverges.
Example 5
Use the comparison test to determine whether the series converges or diverges.
Hint
n < en for n ≥ 1
Answer
This kind-of-sort-of looks like , which converges, so we'll guess the series converges. We need to find a convergent series with bigger terms. To make a fraction bigger we can make the denominator smaller and/or make the numerator bigger.
To make the denominator smaller, use the fact that n < en to replace en with n:
Then make the numerator bigger too. Since cos n is always between -1 and 1,
2 – cos n ≤ 3.
This means
We know that
converges because of the p-test and because multiplication by a constant doesn't affect whether the series diverges. This means the series with smaller terms,
, so we guess it diverges. Since making the denominator smaller makes the fraction bigger,
. Since the series
, so it converges. The comparison test says the series
or like 
? Either way, we guess the series diverges. To show this we need to find a divergent series with smaller terms. If we make the denominator bigger we make the fraction smaller, so replace 
with n:
, which converges, so we'll guess the series converges. We need to find a convergent series with bigger terms. To make a fraction bigger we can make the denominator smaller and/or make the numerator bigger.
