Series Exercises

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.

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.

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.

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.

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,

has to converge.