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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.
- Winter '14