James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 11.2 Series 715
3
Example 11 Find the sum of the series
n−1S ò
nsn 1 1d 1 1 2 nD.
SOLUtion The series o1y2 n is a geometric series with a − 1 2 and r − 1 2 , so
In Example 8 we found that
ò
n−1
ò
n−1
1
1
2 − 2
n 1 2 1 − 1
2
1
nsn 1 1d − 1
So, by Theorem 8, the given series is convergent and
n−1S nD 3
ò
nsn 1 1d 1 1 − 3 ò
2 n−1
1
nsn 1 1d 1 ò
n−1
1
2 n
− 3 ? 1 1 1 − 4 n
Note 4 A finite number of terms doesn’t affect the convergence or divergence of a
series. For instance, suppose that we were able to show that the series
is convergent. Since
ò
n−1
ò
n−4
n
n 3 1 1
n
n 3 1 1 − 1 2 1 2 9 1 3 28 1 ò
n−4
n
n 3 1 1
it follows that the entire series o ǹ−1 nysn 3 1 1d is convergent. Similarly, if it is known
that the series o ǹ−N11 a n converges, then the full series
is also convergent.
ò a n − o N
a n 1
n−1 n−1
ò a n
n−N11
1. (a) What is the difference between a sequence and a series?
(b) What is a convergent series? What is a divergent series?
2. Explain what it means to say that o ǹ−1 a n − 5.
3–4 Calculate the sum of the series o ǹ−1 a n whose partial sums
are given.
3. s n − 2 2 3s0.8d n 4. s n − n 2 2 1
4n 2 1 1
;
7. ò sin n
n−1
s21d
8. ò
n21
n−1 n!
9–14 Find at least 10 partial sums of the series. Graph both the
sequence of terms and the sequence of partial sums on the same
screen. Does it appear that the series is convergent or divergent?
If it is convergent, find the sum. If it is divergent, explain why.
5–8 Calculate the first eight terms of the sequence of partial
sums correct to four decimal places. Does it appear that the
series is convergent or divergent?
1
5. ò
n−1 n 4 1 n 2
6. ò
n−1
1
s 3 n
12
9. ò
n−1 s25d n
11. ò
n−1
n
sn 2 1 4
10. ò cos n
n−1
7
12. ò
n11
n−1 10 n
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