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10. Sequences, Infinite Series, Improper Integrals.

Definition. A function f whose domain is the set of all positive integers 1,2,3.... is called an infinite sequence. The function value f(n) is called the nth term of the sequence.

Definition. A sequence {f(n)} is said to have a limit L if, for every positive number ε, there is another positive number N (which may depend on ε) such that |f(n)-L| [ ε for all n ≥ N. In this case, we say the sequence {f(n)} converges to L and we write , or f(n) → L as n→∞. A sequence which does not converge is called divergent.

Theorem 10.1. A monotonic sequence converges if and only if it is bounded.

Point. Sn = ≥ log (n+1)
Point. = 2 - .

Theorem 10.2. Let ∑an and ∑bn be convergent infinite series of complex terms and let α and β be complex constants. Then the series Σ(α an + β bn) also converges, and its sum is given by the equation = α + β.

Theorem 10.3. If ∑an converges and if ∑bn diverges, then Σ(an…(생략)