PP 12.2 - Series Definition An infinite series is the sum...

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Series
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Definition An infinite series is the sum of the terms of an infinite sequence 1 { } n n a = = 1 n n a Sequence Series = = 1 1 1 1 n n n n = = 1 1 1 sin 1 sin n n n n
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Objective Given an infinite series, to determine if it has a finite sum. How do we find the result of adding infinitely many terms? By constructing a new sequence, called the sequence of partial sums , and determining if it converges or not.
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Example = 1 2 1 n n 2 1 1 = s 4 3 4 1 2 1 2 = + = s 8 7 8 1 4 1 2 1 3 = + + = s 16 15 16 1 8 1 4 1 2 1 4 = + + + = s 32 31 32 1 16 1 8 1 4 1 2 1 5 = + + + + = s Is there any pattern?
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2 1 1 = s 4 3 2 = s 8 7 3 = s 16 15 4 = s 32 31 5 = s n n n s 2 1 2 - = ? convergent 2 1 2 sequence the Is 1 = - n n n 1 2 1 1 lim 2 1 2 lim = - = - n n n n n The limit exists, so the sequence is convergent. 1 2 1 : Hence 1 = = n n
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Definition = = = n i i n n n a s a 1 1 let , series a Given { } . is sum it and converent is series the then , to converges sequence the If 1 1 s a s s n n n n = = = = 1 is, That n n s a Otherwise the series is divergent.
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PP 12.2 - Series Definition An infinite series is the sum...

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