Series Exercises

Answer

Look at the series whose terms are the absolute values.

Since |cos n| is between zero and 1,

By the p-test we know that

converges, and by the comparison test we then know that

converges. The original series converges absolutely.

Answer

The ratio test looks at the absolute value of the ratio between terms:

This is the same thing as the ratio between the absolute values of the terms:

This means when we use the ratio test, we're really checking the convergence of the series

If the ratio test says "yes, the series converges" it actually means "yes, the original series converges absolutely."

For this particular series, we look at

Since the limit is less than 1, the series converges. Since we were really looking at the terms of , the original series converges absolutely.

Answer

This is the classic example of conditional convergence, so we should pay close attention to this answer. We know that  converges by the AST,

and we know by the integral test that

does not. This means the series  converges conditionally.

Answer

(a) There's really nothing to do here. Since  converges and multiplication by a constant doesn't affect convergence,

converges too. We could also say that since  converges, so does

(b) Since ,

Since  can't be less than zero (it could be equal, but not less than),

This is just what we need for the comparison test. We know from (a) that  converges, so by the comparison test

converges.

(c) We know that  and  converge. This means we can take the difference of the series and get a new series that also converges:

This is what we wanted to show, so we're done.