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Home Infinite Series Absolute and Conditional Convergence Theorem 1: Convergence of Absolutly Convergent Series  
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Theorem 1: Convergence of Absolutly Convergent Series
THEOREM 1 Every absolutely convergent series is convergent. That is, if the absolute value series converges, then converges. Discussion This theorem shows that if a positive term series is convergent, then it remains convergent if we make some or all of the terms b_{n} negative, because the new series will still be absolutely convergent. Given an arbitrary series , the theorem shows that exactly one of the following three things can happen: The series is absolutely convergent. The series is conditionally convergent. The series is divergent. PROOF OF THEOREM 1 We use the Sum Rule. Assume converges and let b_{n} = a_{n} + a_{n}. Then a_{n} = b_{n}  a_{n} and (See Figure 9.6.1). Figure 9.6.1 Both and have nonnegative terms. Moreover, converges and b_{n} ≤ 2 a_{n}. By the Comparison Test,
converges. Then using the Sum and Constant Rules, converges.


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