From Positive Series to Ordinary Series

This is actually the expansion of the field of series. In the previous section, we learned about how to handle with most positive series. In this section, we will make good use of the previous tools and make then applicable for almost most ordinary series.

Alternating Series and Its Test

Series in the form ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n+1}{a}_n}$ are called alternating series.

We have only one method to find test the alternating series, which is called the Leibniz's Test. In this test, a series ${\displaystyle \sum a_n}$ satisfies the following three conditions converges:

1. ${\displaystyle a_{n>0}}$
2. ${\displaystyle a_n\ge a_{n+1}}$ for all ${\displaystyle n\ge N}$
3. ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$

With these three conditions True, we can say that a alternating series converges.

Notice that the alternating harmonic series converges while the original harmonic series diverges.

Absolute and Conditional Convergence

We first introduce a theorem:

If ${\displaystyle \sum |a_n|}$ converges, then ${\displaystyle \sum a_n}$ converges.

With this theorem, we always have three cases:

• ${\displaystyle \sum |a_n|}$ converges, which is the absolute convergence.
• ${\displaystyle \sum a_n}$ converges while ${\displaystyle \sum |a_n|}$ diverges, which is the conditional convergence.
• ${\displaystyle \sum a_n}$ diverges.

When facing problems involving ordinary series, we usually need to note if the series is conditional or absolute convergent.

Absolute Ratio Test