## Series with non negative real terms.

We recall here the most important tests for convergence, which will be useful in the next sections. There exist other tests; the interested reader can see them in the Calculus Tutorial.

$\mathbb{C}$


Proposition 7.1.7 (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 7.1.8   Consider the series . This is a series with positive terms.

For , we have . By the integral test (consult the Calculus Tutorial), the series is convergent; thus the given series is convergent.

Proposition 7.1.9 (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 7.1.10   Consider the series , where . For arbitrarily large , we have .

Take . Then:

Thus:

The series is a Riemann series with , thus it is convergent; therefore, by Thm 1.9, the given series is convergent.

Proposition 7.1.11 (Ratio Test; d'Alembert's 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 7.1.12   Take the series with general term . We have:

Thus:

As this limit is more than 1, the given series is divergent.

Example 7.1.13   The series with general term is given. We have:

Thus:

By Thm 1.11, the given series diverges.

Noah Dana-Picard 2007-12-24