# lecture6 - 4 Power Series Sequences and series We say a...

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4. Power Series Sequences and series. We say a sequence s n C converges to s C if, given any ϵ > 0, there exists N N such that | s n s | < ϵ for all n N . The series k =0 z k (we don’t need to start at k = 0) converges if the sequence of partial sums s n = n k =0 z k converges. In this case, the limit of the sequence is called the sum of the series. A series which does not converge is said to be divergent. Absolute convergence. We say that n =0 z n is absolutely convergent if the real series n =0 | z n | is convergent. Exercise. n =0 z n is convergent if and only if n =0 Re z n and n =0 Im z n are both convergent. Suppose n =0 z n is absolutely convergent and write z n = x n + iy n . Then | x n | , | y n | ≤ | z n | so, by the Comparison Test, the real series n =0 x n , n =0 y n are absolutely convergent and hence convergent. Hence (by the exercise) n =0 z n is convergent.

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