The Integral Test

THE INTEGRAL TEST: Suppose is a continuous, positive, decreasing function on and let Then the series is convergent if and only if the improper integral is convergent. In other words:

(a) If is convergent, then is convergent.

(b) If is divergent, then is divergent.

REMARK: Don't use the Integral Test to evaluate series, because in general



EXAMPLES:

1. is divergent, because is continuous, positive, decreasing and is divergent by the -test for improper integrals, since

2. is divergent, because is continuous, positive, decreasing and is divergent by the -test for improper integrals, since

3. is convergent, because is continuous, positive, decreasing and is convergent by the -test for improper integrals, since

REMARK 1: When we use the Integral Test it is not necessary to start the series or the integral at For instance, in testing the series



REMARK 2: It is not necessary that be always decreasing. It has to be ultimately decreasing, that is, decreasing for larger than some number

EXAMPLES: Determine whether the following series converge or diverge.

(a)


(b)


(c)


THEOREM ( -Test): The -series is convergent if and divergent if


REMARK: As before, when we use the -Test it is not necessary to start the series at

EXAMPLES: Determine whether the following series converge or diverge:

(a)


(b)


(c)


(d)



Comparison Tests

THE COMPARISON TEST: Suppose that and are series with positive terms.

(a) If is convergent and for all then is also convergent

(b) If is divergent and for all then is also divergent

EXAMPLES: Use the Comparison Test to determine whether the following series converge or diverge.

(a)


(b)


(c)


(d)


(e)



THE LIMIT COMPARISON TEST
: Suppose that and are series with positive terms. If



where is a finite number and then either both series converge or both diverge.

EXAMPLES: Use the Limit Comparison Test to determine whether the following series converge or diverge:

(a)


(b)


(c)


(d)


(e)