In this case we say that the geometric series

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Unformatted text preview: ries converges, and write ∞ xn = n=0 2 1 . 1−x On the other hand, if |x| > 1, then xN +1 gets larger and larger as N approaches infinity, so limN →∞ xN +1 does not exist as a finite number, and neither does limN →∞ sN . In this case, we say that the geometric series diverges . In summary, the geometric series ∞ xn converges to n=0 1 1−x when |x| < 1, and diverges when |x| > 1. This behaviour, convergence for |x| < some number, and divergences for |x| > that number, is typical of power series: Theorem. For any power series ∞ a0 + a1 (x − x0 ) + a2 (x − x0 ) + · · · = an (x − x0 )n , 2 n=0 there exists R, which is a nonnegative real number or ∞, such that 1. the power series converges when |x − x0 | < R, 2. and the power series diverges when |x − x0 | > R. We call R the radius of convergence . A proof of this theorem is given in more advanced courses on real or complex analysis.1 We have seen that the geometric series ∞ 1 + x + x2 + x3 + · · · = xn n=0 has radius of convergence R = 1. More generally, if b...
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This document was uploaded on 01/12/2014.

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