Chapter 8 Infinite Series,8.3 Infinite Series,8.1 Limits of Sequences of Numbers,8.2 Subsequences, Bounded Sequences, and,8.4 Series of Nonnegative Terms,8.5 Alternating Series, Absolute and Conditional Convergence,8.6 Power Series,8.7 Taylor and Maclaurin Series,8.8 Applications of Power Series,8.4,Series of Nonnegative Terms (正项级数),Given a series,We take care of two questions:,1. Does the series converge?,2. If it converges, what is its sum?,Let us begin with a particular kind of series,According to Theorem 5(P621),Series of nonnegative terms,for all n.,Example 1. Show that the following series converges?,Solution.,Let,Consider the partial sum,Example 2. The following converges ?,continues, positive, decreasing.,Solution.,Solution.,Example 3. The p-series,Example 4. The following converges ?,continues, positive, decreasing.,Solution.,比 较 判 别 法,Example 1. Applying the comparison test,diverges,Solution.,Example 2. Applying the comparison test,We consider,Solution.,Example 3. Applying the comparison test,We consider,Solution.,比较判别法的极限形式,Example 1. Applying the limit comparison test,diverges.,Solution.,Example 2. Applying the limit comparison test,diverges.,Solution.,Example 3. Applying the limit comparison test,converges.,Solution.,比 值 判 别 法,For,Example 1. Applying the ratio test,Solution.,Example 2. Applying the ratio test,Solution.,Example 3. Applying the ratio test,Solution.,根 值 判 别 法,Example 1. Applying the root test,Solution.,Example 2. Applying the root test,Solution.,Exercises,P649 4, 5, 10, 11, 13, 16, 18, 19, 22, 25, 27, 29, 30, 34.,