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1 AP Calculus BC Review: Sequences, Infinite Series, and Convergence Sequences A sequence {๐๐ } is a function whose domain is the set of positive integers. The functional values a1, a2, a3, . . . an are called the terms of the sequence. The number an is called the nth term of the sequence. A sequence is either convergent (if the sequence has a limit, i.e. it approaches a specific number) or divergent (if the sequence does not approach a specific limit) A sequence {๐๐ } has the limit L DEF 1 ๐๐๐ ๐๐ = ๐ฟ ๐โโ ๐๐ ๐๐ โ ๐ฟ ๐๐ ๐ โโ if we can make the terms ๐๐ as close to ๐ฟ as we like by taking ๐ sufficiently large. If ๐๐๐ ๐๐ exists , we say the sequence is convergent. Otherwise, we say the sequence is divergent. ๐โโ DEF 2 ๐๐๐ ๐๐ = โ ๐โโ means that for every positive number ๐ there is an integer N such that ๐๐ > ๐ whenever n > N It is a special type of divergence. We say {๐๐ } diverges to โ . โ Sequence {โ1, 1, โ1, 1, โ1, 1, โ1, โฆ } is divergent, but doesnโt diverges to โ . โ Sequence ๐๐ = ๐2 + 3 is divergent too, and it does diverges to โ . 2๐ โ 5 If {๐๐ } and {๐๐ } are convergent sequences and c is a constant, then THM 1 ๐๐๐ (๐๐ ± ๐๐ ) = ๐๐๐ ๐๐ ± ๐๐๐ ๐๐ ๐โโ ๐โโ ๐โโ ๐๐๐ ๐๐๐ = ๐ ๐๐๐ ๐๐ ๐โโ ๐โโ ๐๐๐ (๐๐ ๐๐ ) = ๐๐๐ ๐๐ โ ๐๐๐ ๐๐ ๐โโ ๐๐๐ ๐โโ ๐โโ ๐โโ ๐ ๐๐๐ (๐๐ )๐ = (๐๐๐ ๐๐ ) ๐โโ ๐โโ ๐๐๐ ๐๐ ๐๐ = ๐โโ ๐๐ ๐ ๐๐๐ ๐๐ ๐โโ ๐๐๐ ๐ make the terms 1 ๐๐ as close๐โโ if we can to ๐ฟ1 as we like by 1taking ๐ sufficiently large. โ ๐๐๐ = ๐๐๐ = = = 1 ๐โโ ๐ + 1 ๐โโ 1 + 1โ 1 + 0 ๐๐๐ 1 + ๐๐๐ 1โ ๐ ๐ If ๐๐๐ ๐๐ exists , we say the sequence is convergent. ๐โโ ๐โโ Otherwise, we say the sequence is divergent. ๐โโ THM 2 - ๐๐ก๐ ๐๐ช๐ฎ๐๐๐ณ๐ ๐ญ๐ก๐ฆ If ๐๐ โค ๐๐ โค ๐๐ for ๐ โฅ ๐0 and ๐๐๐ ๐๐ = ๐๐๐ ๐๐ = ๐ฟ , then ๐๐๐ ๐๐ = ๐ฟ ๐โโ โ ๐๐๐ ๐! =? ๐๐ ๐๐ = 1 โ 2 โ 3 โโโ ๐ 1 1 โ 2 โ 3 โโโ ๐ = ( ) ๐ โ ๐ โ ๐ โโโ ๐ ๐ ๐ โ ๐ โ ๐ โโโ ๐ ๐โโ ๐๐๐ 0 = 0 ๐โโ ๐๐๐ 1 ๐โโ ๐ = 0 (๐๐ป๐ 2) โน ๐โโ ๐โโ 1 โ 2 โ 3 โโโ ๐ โค1 โน ๐ โ ๐ โ ๐ โโโ ๐ ๐๐๐ ๐! ๐โโ ๐๐ =0 0 < ๐๐ โค 1 ๐ 2 If lim ๐(๐ฅ) = ๐ฟ ๐๐๐ ๐(๐) = ๐๐ , when n is an integer, then ๐๐๐ ๐๐ = ๐ฟ THM 3 ๐ฅโโ โ ๐๐๐ ๐โโ ๐โโ ๐๐ ๐ =? ๐ ( โ ) โ l'Hospital's rule cannot be applied to sequences but to function of real variable. Introducing ๐(๐ฅ) = ๐๐ ๐ฅโ๐ฅ we can use it now. ๐๐๐ ๐ฅโโ 1โ ๐๐ ๐ฅ ๐ฅ = ๐๐๐ = 0 ๐โโ 1 ๐ฅ โ ๐๐๐ ๐โโ 3๐ =? ๐ 5๐ ( (๐๐ป๐ 3) โน ๐๐๐ ๐โโ ๐๐ ๐ =0 โน ๐ โ ) โ Sequence is obviously approaching 0 as n approaches infinity. Letโs apply THM 3 anyway. ๐(๐ฅ) = 3๐ฅ ๐ 5๐ฅ 3๐ฅ โ = ( )= 5๐ฅ ๐ฅโโ ๐ โ lim ๐(๐ฅ) = lim ๐ฅโโ โ Show that the sequence { lim ๐(๐ฅ) = lim ๐ฅโโ ๐ฅโโ 1+2๐3 ๐3 lim ๐ฅโโ 3 =0 โน 5๐ 5๐ฅ ๐๐๐ ๐โโ 3๐ =0 ๐ 5๐ } converges 1 + 2๐ฅ 3 โ = ( )= 3 ๐ฅ โ lim ๐ฅโโ 1/๐ฅ 3 + 2 =2 โน 1 1 + 2๐3 = 2 ๐โโ ๐3 lim We will often have to find a formula for the general, or nth, term of a sequence. Look at the next example: โ If the first four terms of a sequence {an} are 1 , 9 27 81 , , 7 11 15 a) find a formula for the nth term of the sequence b) determine whether the sequence converges or diverges a) sequence is: 31 32 33 34 3 nth term is: ๐๐ = ๐) ๐(๐ฅ) = THM 4 , 7 3๐ , 11 , 15 , โฏ , = 31 32 33 4๐โ1 3๐ฅ 4๐ฅ โ 1 3๐ฅ โ 3 ๐ฅ ๐๐ 3 = ( ) = lim = โ ๐ฅโโ 4๐ฅ โ 1 โ 4 lim ๐ฅโโ If lim |๐๐ | = 0 , then ๐๐๐ ๐๐ = 0 ๐ฅโโ โ ๐๐๐ ๐โโ ๐๐๐ | ๐โโ ๐โโ (โ1)๐ =? ๐ (โ1)๐ 1 | = ๐๐๐ =0 ๐โโ ๐ ๐ (๐๐ป๐ 4) โน ๐๐๐ ๐โโ (โ1)๐ =0 ๐ โ For what values of ๐ is the sequence {๐ ๐ } convergent? ๐=0 ๐=1 0<๐<1 โ1 < ๐ < 0 34 , , , ,โฏ 4(1)โ1 4(2)โ1 4(3)โ1 4(4)โ1 ๐๐๐ 0๐ = 0 ๐โโ ๐๐๐ 1๐ = 1 ๐โโ ๐๐๐ ๐ ๐ = 0 ๐โโ ๐๐๐ |๐ ๐ | = 0 ๐โโ (๐๐ป๐ 4) โน ๐๐๐ ๐ ๐ = 0 ๐โโ {๐ ๐ } converges for โ 1 < ๐ < 1 and diverges for |๐| > 1 โน the sequence diverges. 3 DEF 3 โ MONOTONIC SEQUENCE is a sequence {๐๐ } such that either it is increasing, that is ๐๐+1 > ๐๐ for every ๐ โฅ 1 , or decreasing, that is ๐๐+1 < ๐๐ for every ๐ โฅ 1 . ๐ is decreasing. ๐2 + 1 ๐+1 ๐ โน < 2 โน (๐ + 1)(๐2 + 1) < ๐[(๐ + 1)2 + 1] (๐ + 1)2 + 1 ๐ +1 โ Show that sequence ๐๐ = ๐๐+1 < ๐๐ โน ๐3 + ๐2 + ๐ + 1 < ๐3 + 2๐2 + 2 ๐ โน 1 < ๐2 + ๐ For ๐ โฅ 1 inequality [1 < ๐2 + ๐] is true, so ๐๐ = ๐ ๐2 +1 is decreasing. A sequence {๐๐ } is bounded above if there is a number ๐ such that DEF 4 ๐๐ โค ๐ for every ๐ โฅ 1 A sequence {๐๐ } is bounded below if there is a number ๐ such that ๐๐ โฅ ๐ for every ๐ โฅ 1 A sequence {๐๐ } is bounded if it is bounded above and below. THM 5 โ MONOTONIC SEQUENCE THEOREM โ A bounded monotonic increasing sequence is convergent (it converges to its greatest lower bound). โ A bounded monotonic decreasing sequence is convergent (it converges to its greatest lower bound). โ A convergent sequence is bound. โ There are sequences which are convergent without being monotonic. Sequences: 1 1 1 2 3 4 {โ1, , โ , , โฆ } and 1 { , 1 1 1 , , 2 22, 3 32 , โฆ } both converge to 0. Mathematical induction is a method of mathematical proof typically used to establish that a given statement P(n) is true for all natural numbers 1. step: verify that P(n) is true for n = i (i is usually 1) 2. step (inductive step): assuming that there is a ๐ โฅ 1, for which P(k) is true, then 3. step: prove that, P(k+1) is true. Since you have verified P(i), it follows from the inductive step that P(i + 1) is true, and hence, P(i + 2) is true, and hence P(i + 3) is true, and so on. In this way the theorem has been proved. The Fibonacci numbers are recursively defined as ๐1 = 1, ๐2 = 1, ๐๐ = ๐๐โ1 + ๐๐โ2 , ๐ > 2 Show that the sequence of Fibonacci numbers {1, 1, 2, 3, 5, 8, 13, 21, ...} does not converge. We will show by induction that the sequence of Fibonacci numbers is unbounded. If that is true, then the sequence can not converge, because every convergent sequence must be bounded. We will show that the n-th term of that sequence is greater or equal to n, at least for n > 4. Property P(n): ๐๐ โฅ ๐ for all ๐ > 4 1. Check the lowest term: ๐5 = ๐4 + ๐3 = 5 โฅ 5 is true 2. Assume: ๐๐ โฅ ๐ for all ๐ > 4 3. Prove: ๐๐+1 โฅ ๐ + 1 ๐๐+1 = ๐๐ + ๐๐โ1 โฅ ๐ + ๐ โฅ ๐ + 1 โฅ ๐ , Hence, by induction the Fibonacci numbers are unbounded and the sequence can not converge.