Presentation on theme: "A second type of series has a simple arithmetic test for convergence or divergence. A series of the form is a p-series, where p is a positive constant."— Presentation transcript:
A second type of series has a simple arithmetic test for convergence or divergence. A series of the form is a p-series, where p is a positive constant. For p = 1, the series is the harmonic series. p-Series and Harmonic Series p-series Harmonic series
A general harmonic series of the form In music, strings of the same material, diameter, and tension, whose lengths form a harmonic series, produce harmonic tones. p-Series and Harmonic Series
For example, in the following pairs, the second series cannot be tested by the same convergence test as the first series even though it is similar to the first. Direct Comparison Test
Often a given series closely resembles a p -series or a geometric series, yet you cannot establish the term- by-term comparison necessary to apply the Direct Comparison Test. Under these circumstances you may be able to apply a second comparison test, called the Limit Comparison Test.
For a convergent alternating series, the partial sum S N can be a useful approximation for the sum S of the series. The error involved in using S S N is the remainder RN = S – S N. Alternating Series Remainder
Absolute and Conditional Convergence
The converse of Theorem 9.16 is not true. For instance, the alternating harmonic series converges by the Alternating Series Test. Yet the harmonic series diverges. This type of convergence is called conditional. Absolute and Conditional Convergence