SECTION 12.4 The Limit of a Sequence; Infinite Series 891 To define the sum of an infinite series ∑ = ∞ a , k k 1 we make use of the sequence { } Sn defined by S a S a a a S a a a a k k n n k n k 1 1 2 1 2 1 2 1 2 1 ∑ ∑ = = + = ⋅ ⋅ ⋅ = + + + = = = The sequence Sn { } is called the sequence of partial sums of the series a . k k 1 ∑ = ∞ For example, consider again the series ∑ =+ + + +=++++ = ∞ 1 2 1 2 1 2 1 2 1 2 1 2 1 4 1 8 1 16 k k 1 2 3 4 As it turns out, the partial sums can be rewritten as 1 minus a power of 1 2 , as follows: = = − = + = = − = − = + + = = − = − = = − = − = = − = − S S S S S S 1 2 1 1 2 1 2 1 4 3 4 1 1 4 1 1 2 1 2 1 4 1 8 7 8 1 1 8 1 1 2 15 16 1 1 16 1 1 2 31 32 1 1 2 1 1 2 n n 1 2 2 3 3 4 4 5 5 The n th partial sum is = − S 1 1 2 . n n As n gets larger and larger, the sequence Sn { } of partial sums approaches a fixed value or limit . That is, ( ) = − = →∞ →∞ S lim lim 1 1 2 1 n n n n We call this limit the sum of the series and write 1 2 1 2 1 4 1 8 1 16 1 k k 1 ∑ = + + + + = = ∞ DEFINITIONS Converge and Diverge If the sequence { } Sn of partial sums of an infinite series ∑ = ∞ a k k 1 has a limit S, then the series converges and is said to have the sum S . That is, if = →∞ S S lim , n n then ∑ = + + + + = = ∞ a a a a S k k n 1 1 2 An infinite series diverges if the sequence of partial sums diverges.
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