This document provides an overview of various tests that can be used to determine whether an infinite series converges or diverges, including the comparison test, harmonic series test, Kummer's test, D'Alembert's ratio test, Gauss test, Cauchy root test, and condensation test. These tests examine properties of the sequence of terms in an infinite series or its sequence of partial sums to establish whether the series converges absolutely or conditionally, or diverges.

1. -K.Anitha M.Sc.,M.Phil.,
-A.Sujatha M.Sc.,M.Phil.,
Sequences and Series

2. Series
 Let (an ) be a sequence of real numbers . Then
a1+a2+……an+…is called an infinite series of real
numbers.
 (sn ) is called the sequence of partial sumsof the
given series.The series converges or diverges
according as (sn ) converges or diverges.

3. Comparison Test

4. Harmonic Series

5. Example

6. Kummer’s Test

7. D’Alembert’s Ratio Test

8. Gauss Test

9. Cauchy Root Test

10. Condensation Test