Series (12.2) A series is the sum of a sequence. If {an } is a sequence, we may add it up as
a1 + a2 + a3 + + an + =
n=1
an .
What does it mean to add up infinitely many numbers? We define the n-th partial sum sn of the series by
n
sn = a1 + a2 + a3 + + an =
i=1
ai .
We say that the series converges if the sequence {sn } of partial sums converges; otherwise,
n
we say the series diverges. That is, the sum of the series is lim sn = lim
n
n
ai , if that
i=1
limit exists. Example. Consider 1 1 1 1 1 = + + + + n + . The first four partial sums are 2n 2 4 8 2 n=1 1 3 7 15 1 . It turns out sn = 1 - n . So the sum of the series s1 = , s2 = , s3 = and s4 = 2 4 8 16 2 1 1 is lim sn = lim 1 - n = 1 - 0 = 1. Why is sn = 1 - n ? Here's a simple way of n n 2 2 verifying this: Write 1 1 1 1 + + + + n 2 4 8 2 1 1 1 so 2sn = 1 + + + + n-1 2 4 2 1 2sn - sn = 1 - n (things cancel) 2 1 so sn = 1 - n . 2 sn =
so subtract to get
Example. A geometric series is any series of the form
arn = a + ar + ar2 + ar3 + ,
n=0
where a and r are constants. We call a the initial term and r the ratio of the series. The partial sums of this series turn out to be
n
sn =
i=1
ari = 1
a(1 - rn+1 ) . 1-r
This is verified using the same trick as the example above. (Subtract sn from rsn and things cancel.) The limit lim sn = lim a(1 - rn+1 ) converges exactly when lim rn converges; this n n n 1-r a a(1 - rn+1 ) = . Therefore, the occurs when |r| < 1. In that case, the limit is lim n 1-r 1-r a arn converges to geometric series if |r| < 1 but diverges otherwise. 1-r n=0
n
1 3 1 1 3 = . Since Example. Try a = 3 and r = . This is the series < 1, n 5 5 5 5 n=0 n=0 3 15 a = . Try a few partial sums on a calculator and the series converges to 1 = 1-r 4 1- 5 see how rapidly they converge to the exact sum 3.75. For example, s5 = 3 + 3 3 3 3 3 + + + + = 3.74976. 5 25 125 625 3125
We can also find s5 using the formula for the n-th partial sum of a geometric series: 3 1 - ( 1 )6 a(1 - r6 ) 11718 5 = . = s5 = 1 1-r 3125 1- 5 Example. Repeating decimals can be written as geometric series. For example, 0.454545 = 45 45 45 + + + 100 10000 1000000 45 45 45 + = + + . 2 100 100 1003
This is a geometric series with a = 45/100 and r = 1/100. So it converges to
45 45 45 5 45 a 45 + = 100 1 = = . + + = 2 3 100 100 100 1-r 99 11 1 - 100
1 1 1 1 1 Example. A "telescoping" series: + + + + + = . 12 23 34 n(n + 1) n(n + 1) n=1 We can find an exact formula for the n-th partial sum. We use partial fractions to rewrite the n-th term: A B 1 = + n(n + 1) n n+1 which leads to 1 = A(n + 1) + Bn or 1 = (A + B)n + A. This forces A + B = 0 and A = 1, so B = -1. Therefore, 1 1 1 = - n(n + 1) n n+1 2
Now the n-th partial sum becomes sn = 1 1 1 1 + + + + 12 23 34 n(n + 1) 1 1 1 1 1 1 = - + - + - + 1 2 2 3 3 4 1 =1- n+1
1 1 - n-1 n
+
1 1 - n n+1
For example, s4 = 1 1 1 1 + + + 12 23 34 45 1 1 1 1 1 1 - + - + - = 1 2 2 3 3 4 1 4 =1- = . 5 5
+
1 1 - 4 5
Now to find the sum of the series, we compute 1 1 = 1 - 0 = 1. So it adds up to 1. = lim sn = lim 1 - n n(n + 1) n n+1 n=1
Example. A simple divergent series:
(-1)n = -1 + 1 - 1 + 1 - 1 + 1 - 1 + . The
n=1 n
n-th partial sum is sn = 0 if n is even, but it is -1 if n is odd. So lim sn does not exist.
Example. Another divergent series:
n=0
2n . The n-th partial sum is sn = 2n+1 - 1
(using the formula for the partial sum of a geometric series, with a = 1 and r = 2). Now lim sn = . The famous mathematician Leonard Euler speculated (back in the 1700s) n a that there was a connection between negative numbers and infinity since the formula 1-r 1 is = -1 so -1 = . This is of course nonsense we aren't allowed to use the formula 1-2 unless |r| < 1.
Theorem. If
n=1
an converges, then lim an = 0.
n
Proof Suppose s =
n=1 n
an , so lim sn = s. Then lim sn-1 = s too, so lim sn =
n n n n n
lim (sn-1 + an ), which forces s = s + lim an , so lim an = 0. So we're done. 3
This theorem gives a quick way of verifying some series diverge. For example, diverges since lim
n 2n + 1 n=1
1 n = = 0. This is called the divergent series test. But n 2n + 1 2 WARNING, you cannot use this test to show series converge. For example, you might 1 1 converges since 0, but in fact, it diverges. suppose that the "harmonic series" n n n=1
Theorem. Suppose
n=1
an = a and
n=1
bn = b. Then
n=1
can = ca and also
(an +bn ) =
n=1
a + b. We can use this theorem to sum some series up. For example,
n=1
5 -4 n(n + 1)
3 5n
= 5(1) - 4
15 4
= 5 - 15 = -10.
4