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lim S N

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provided that this limit exists

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Discrete Mathematics Demystified

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132 Some Examples

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EXAMPLE 133 Does the series

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2 j converge If so, to what limit

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Solution: We have

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SN =

2 j = 2 1 + 2 2 + 2 3 + + 2 N

= 2 1 + (2 1 2 2 ) + (2 2 2 3 ) + (2 3 2 4 ) + + (2 (N 1) 2 N ) (131)

Notice that S N is a nite sum so that it is correct to use the associative law of addition We conclude that S N = 2 1 + 2 1 + ( 2 2 + 2 2 ) + ( 2 3 + 2 3 ) + + ( 2 (N 1) + 2 (N 1) ) 2 N = 1 2 N We may now pass to the limit:

N

lim S N = lim (1 2 N ) = 1 lim 2 N = 1 0 = 1

N N

By de nition, we say that the series

j=1

2 j

converges to 1 Refer to Fig 131

Figure 131 Convergence of the series

2 j

Series

Insight: Do not worry about how one thinks of the trick that was used in Example 133 The purpose of this chapter is to teach you all the techniques that you will need in order to study series We will learn them gradually, beginning in the next section For now, just concentrate on learning what it means for a series to converge or diverge EXAMPLE 134 Discuss convergence for the series

j=1

( 1) j+1 = 1 1 + 1 1 + 1

Solution: If N is odd then SN = 1 1 + 1 1 + + 1 = (1 1) + (1 1) + + (1 1) + 1 =1 However, if N is even then SN = 1 1 + 1 1 + 1 = (1 1) + (1 1) + + (1 1) =0 Therefore the sequence {S N } =1 of partial sums is just the sequence N 1, 0, 1, 0, 1, 0, which does not converge We conclude that the series itself does not converge Insight: Example 134 explains away the apparent contradiction that we encountered in Example 131 The lesson is that we should not attempt to perform arithmetic operations on series that do not converge WARNING: It is easy to become confused at this point about the difference between a sequence and a series Remember that a sequence is a list of numbers while a series is a sum of numbers However, we study a series by looking at its sequence of partial sums

Discrete Mathematics Demystified

Sometimes there are tricks involved in seeing that a series converges: EXAMPLE 135 Discuss convergence for the series

j=1

1 j ( j + 1)

Solution: If you use a calculator to compute some partial sums (up to S20 , for instance) then you will probably be convinced that the series converges Here is a way to get this conclusion mathematically: We write SN = = 1 1 1 1 + + + + 1 2 2 3 3 4 N (N + 1) 1 1 1 2 + 1 1 2 3 + 1 1 3 4 + + 1 1 N N +1

Almost everything cancels out and we have SN = 1 1 N +1

Clearly lim N S N = 1 Therefore the series

j=1

1 j ( j + 1)

converges to 1 Insight: Sometimes it is convenient to begin a series at an index other than j = 1 An example is

j=3

j ln( j 1)

Obviously it would not do to begin this series at j = 1 because ln 0 is unde ned We cannot begin at j = 2 because ln 1 = 0 So we begin at j = 3 Of course this

series is equal to

Series

3 4 + + ln 2 ln 3 But notice this: the partial sum S1 is understood to be 0 because summing from 3 to 1 makes no sense The partial sum S2 is zero for a similar reason The rst nontrivial partial sum is S3 = Indeed, for N 3 we have SN = 3 4 N + + + ln 2 ln 3 ln(N 1) 3 ln 2