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11.1 Infinite Series

11.1   Definition (Series operator.) If is a complex sequence, we define a new sequence by or We use variations, such as  is actually a function that maps complex sequences to complex sequences. We call the series corresponding to .

11.2   Remark. If are complex sequences and , then and since for all , and 11.3   Examples. If is a geometric sequence, then is a sequence we have been calling a geometric series. If , then is the sequence for that we studied in the last chapter.

11.4   Definition (Summable sequence.) A complex sequence is summable if and only if the series is convergent. If is summable, we denote by . We call the sum of the series .

11.5   Example. If and , then .

11.6   Example (Harmonic series.) The series is called the harmonic series, and is denoted by . Thus We will show that diverges; i.e., the sequence is not summable. For all , we have From the relation , we have and (by induction), Hence, is not bounded, and thus diverges; i.e., is not summable.

11.7   Theorem (Sum theorem for series.) Let be summable sequences and let . Then and are summable, and If is not summable, and , then is not summable.

Proof: The proof is left to you.

11.8   Exercise. Let be summable sequences. Show that is summable and that 11.9   Example. The product of two summable sequences is not necessarily summable. If then This is a null sequence, so is summable and . However, so . Thus is unbounded and hence is not summable.

11.10   Theorem. Every summable sequence is a null sequence. [The converse is not true. The harmonic series provides a counterexample.]

Proof: Let be a summable sequence. Then converges to a limit , and by the translation theorem also. Hence i.e., and it follows that is a null sequence.     Next: 11.2 Convergence Tests Up: 11. Infinite Series Previous: 11. Infinite Series   Index