L16 Series Part I - series tend to 0, determine if the...

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Lecture 16 : Series (I) What is meant by the in±nite series ? a 1 + a 2 + a 3 + ±±± + a n + ±±± = 1 X n =1 a n ? Consider the sequence of partial sums: s 1 = a 1 s 2 = a 1 + a 2 s 3 = a 1 + a 2 + a 3 . . . s n = a 1 + a 2 + ::: + a n If the sequence of partial sums f s n g con- verges, then we say that the series a 1 + a 2 + a 3 + ±±± + a n + ±±± converges , and we de±ne 1 X n =1 a n = a 1 + a 2 + ±±± + a n + ±±± = lim n !1 s n = s 1
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That is, If lim n !1 s n = s , then 1 X n =1 a n = s If f s n g diverges, we say the series 1 X n =1 a n diverges. Key Questions about series: 1. Does the series converge? 2. If the series converges, what is the sum? 2
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Geometric Series The Geometric series 1 X n =0 ar n = a + ar + ar 2 + :::; a 6 = 0 is convergent if j r j < 1 and its sum is 1 X n =0 ar n = a 1 ± r ; a =±rst term. If j r j ² 1 the geometric series is divergent. Theorem: If the series 1 X n =1 a n is convergent, then lim n ± > 1 a n = 0 : Test for Divergence: If lim n ± > 1 a n 6 = 0 or does not exist, then the series is divergent. Warning : If we ±nd that lim n ± > 1 a n = 0, we still know nothing about the convergence or diver- gence of the series. 3
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For Example: The terms of both the following
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Unformatted text preview: series tend to 0, determine if the series converge: 1. 1 X n =1 (3 = 4) n 2. Harmonic Series 1 X n =1 1 n 4 Determine if the series converges or diverges. If converges, nd the sum. ex. 1 X n =1 n 2 5 n 2 + 4 ex. 1 X n =1 ( 1) n ex. 5 10 3 + 20 9 40 27 :::: 5 ex. 1 X n =1 2 2 n 3 1 n ex. 1 X n =1 e n n 2 6 ex. 1 X n =1 1 + 1 n n 7 NYTI: 1 X n =1 (1 = 2) n 1 (Geometric, 2) 1 X n =3 ( 5 = 4) n (Geo., Diverges) 1 X n =1 5 4 n (Geo., 5 3 ) 1 X n =1 e n 3 n 1 (Geo., 3 e 3 e ) 1 X n =1 n p 2 (TFD) Tonights homework: work out series problems: geometric series, harmonic series, divergent series due to test for divergent. 8...
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L16 Series Part I - series tend to 0, determine if the...

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