# Series with non negative terms.

Proposition 9.3.1 (Integral test)   Let be a sequence of positive real numbers. We suppose that , where is a positive, continuous, decreasing function of the real unknown on an interval of the form , where .

Then the series and the improper integral either both converge or both diverge.

Example 9.3.2   Consider the series . We have:
• , where and is a continuous, non-negative, decreasing function on .
• Moreover:

Thus the given series is convergent.

Proposition 9.3.3 (Direct Comparison)   Let be a series of real numbers, such that .
1. Suppose that there exists a convergent series such that, for a given natural number , . Then the series is convergent.
2. Suppose that there exists a divergent series such that, for a given natural number , . then the series is divergent.

Example 9.3.4   Consider the series . This is a series with positive terms.

For , we have . By the integral test (v.s. Thm 3.1 and example 3.2), the series is convergent; thus the given series is convergent.

Proposition 9.3.5 (Limit Comparison)   Let and be two series of real numbers, such that and .
1. If there exists a positive real number L such that , then the two series either are both convergent or are both divergent.
2. If and is convergent, then is convergent.
3. If and is divergent, then is divergent.

Example 9.3.6   Consider the series , where . For arbitrarily large , we have .

Take . Then:

Thus:

By Thm 2.2 and Thm 9.9, the series is convergent; therefore, by Thm 3.5, the given series is convergent.

Proposition 9.3.7 (Ratio Test; d'Alembert's Test)   Let such that . Suppsoe that:

Then:
• If , the series is convergent.
• If or , the series is divergent.
• If , the test gives no conclusion.

Example 9.3.8   Take the series with general term . We have:

Thus:

As this limit is less than 1, the given series is convergent.

Example 9.3.9   The series with general term is given. We have:

Thus:

By Thm 3.7, the given series diverges.

Proposition 9.3.10 (Root Test)   Let such that . Suppose that:

Then:
• If , the series is convergent.
• If or , the series is divergent.
• If , the test gives no conclusion.

Example 9.3.11   Consider the series , where . This is obviously a series with positive terms.

We have:

Therefore:

By Thm 3.10, the given series is convergent.

Noah Dana-Picard 2007-12-28