![]() Since the length of the interval is 2, the radius of convergence is 1. For example, since the series ∑ n = 0 ∞ x n ∑ n = 0 ∞ x n converges for all values x in the interval ( −1, 1 ) ( −1, 1 ) and diverges for all values x such that | x | ≥ 1, | x | ≥ 1, the interval of convergence of this series is ( −1, 1 ). The value R is called the radius of convergence. Since the series diverges for all values x where | x − a | > R, | x − a | > R, the length of the interval is 2 R, and therefore, the radius of the interval is R. The set of values x for which the series ∑ n = 0 ∞ c n ( x − a ) n ∑ n = 0 ∞ c n ( x − a ) n converges is known as the interval of convergence. The series may converge or diverge at the values x where | x − a | = R. Therefore, the series converges for all x such that | x | 0, R > 0, and diverges for all x such that | x − a | > R. Suppose that the set S =, the number R > 0. WritingĪnd let S be the set of real numbers for which the series converges. ![]() Therefore, there exists an integer N such that | c n d n | ≤ 1 | c n d n | ≤ 1 for all n ≥ N. Since ∑ n = 0 ∞ c n d n ∑ n = 0 ∞ c n d n converges, the nth term c n d n → 0 c n d n → 0 as n → ∞. If there exists a real number d ≠ 0 d ≠ 0 such that ∑ n = 0 ∞ c n d n ∑ n = 0 ∞ c n d n converges, then the series ∑ n = 0 ∞ c n x n ∑ n = 0 ∞ c n x n converges absolutely for all x such that | x | < | d |. ![]() ) We must first prove the following fact: (For a series centered at a value of a other than zero, the result follows by letting y = x − a y = x − a and considering the series ∑ n = 1 ∞ c n y n. Suppose that the power series is centered at a = 0. At the values x where | x − a | = R, | x − a | = R, the series may converge or diverge. There exists a real number R > 0 R > 0 such that the series converges if | x − a | R.The series converges for all real numbers x.The series converges at x = a x = a and diverges for all x ≠ a.The series satisfies exactly one of the following properties: We now summarize these three possibilities for a general power series.Ĭonsider the power series ∑ n = 0 ∞ c n ( x − a ) n. For example, the geometric series ∑ n = 0 ∞ x n ∑ n = 0 ∞ x n converges for all x in the interval ( −1, 1 ), ( −1, 1 ), but diverges for all x outside that interval. In that case, the power series either converges for all real numbers x or converges for all x in a finite interval. Most power series, however, converge for more than one value of x. Some power series converge only at that value of x. Therefore, a power series always converges at its center. For a power series centered at x = a, x = a, the value of the series at x = a x = a is given by c 0. Since the terms in a power series involve a variable x, the series may converge for certain values of x and diverge for other values of x.
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