Chapter 4(A) - Chapter 4 Sequences & Series Overview n...

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Chapter 4 Sequences & Series
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Overview n Infinite Sequences q Limits of Sequences q Sequences & Functions n Infinite Series q Partial Sums q Geometric Series q Ratio Test
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Overview n Power Series q Convergence of Power Series q Radius of Convergence q Differentiation and Integration of Power Series n Taylor Series q Definition q Taylor Polynomials q An Application of Taylor Polynomials
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Infinite Sequences
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Infinite Sequences n A sequence of real numbers: 12 , , n a aa LL We use { } to denote an infinite sequence n a : general term of the sequence n a
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Infinite Sequences When we look at an infinite sequence { }, we are looking at an infinite list of numbers: n a 12 , , n a aa LL
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Infinite Sequences - Examples 1 (ii) n a n = (i) 1 n an =- 0, 1, 2, , 1, n - LL 123 , , , n aa 1 11 1, , , 23 n
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Infinite Sequences - Examples 1 1 (iii) ( 1) n n a n +  =-   1 1 11 1, , , 2 3 45 -- L 1 (ii) n a n = 1 1, , , 23 n LL Alternate sign
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Infinite Sequences - Examples Alternate sign 1 ( 1) n + - gives , , +, , +, + -- L ( n - gives , , , +, , +, -+ L
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Infinite Sequences - Examples 1 (iv) n n a n - = 123 0, , , , 234 L 1 1 1 1 , -- L 1 (v) ( 1) n n a + =-
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Infinite Sequences - Examples (vi) n ak = k, k, k, k, L (where is a constant) k constant sequence with every term having value k
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Limits of Sequences A number is called th e of a sequence { }, if for sufficiently large , we can get as close as we want to a number . n n La na L limit lim n n aL →∞ = n We write or
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Limits of Sequences A number is called th e of a sequence { }, if for sufficiently large , we can get as close as we want to a number . n n La na L limit lim n n aL →∞ = n or When we look at the limit of a sequence { }, we are interested to know what happen to the value of when is sufficiently large. n n a an
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Limits of Sequences . is , it if }, { of limit The unique exists n a If li m , we say that { } is and { } to . nn n n a La aL →∞ = convergent converges . is } { say that we , lim If divergent exist not does n n n a a →∞
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Limits of Sequences - Examples Divergent (i) 1 n an =- 0, 1, 2, , 1, n - LL When we look at the limit of a sequence { }, we are interested to know what happen to the value of when is sufficiently large. n n a A number is called th e of a sequence { }, if for sufficiently large , we can get as close as we want to a number .
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Chapter 4(A) - Chapter 4 Sequences & Series Overview n...

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