Discrete Mathematics Demystified

CHAPTER 13 Series 299
In each of Exercises 3 and 4, use a calculator to determine some partial sums and
guess whether the series converges. If you can (for the cases of convergence), state
what the sum should be.
3. ∞
j=1
j
j+1
4. ∞ 1 1
j=1 ( − j j+1 )
Each of the series in Exercises 5 and 6 converges. Explain why. Can you find the
sum?
5. ∞ 2
j=4 ( j−2)( j−3)
6. ∞ j=1 2− j cos( jπ)
In Exercises 7 and 8, find the explicit sum of the series.
7. ∞ j
j=0
8−
8. ∞ j
j=0 (3/7)
In Exercises 9 and 10, calculate each of the sums by using the formula for the partial
sum of a geometric series which appears in the text.
9. 6 j
j=0
3−
10. 12 j+1
j=3 11
In Exercises 11 and 12, use the comparison tests to determine whether the series
converges or diverges.
11. ∞ sin
j=1
2 j
j 2
12. ∞
j=1
j 2
j 3 +1
In Exercise 13 , test the given series for convergence or divergence by using the
Ratio Test. If the test gives no information, then say so explicitly.
13. ∞
j=1 1
j!
In Exercise 14, test the given series for convergence or divergence by using the
Root Test. If the test gives no information, then say so explicitly.
j
14. ∞
j=1
3 j
j+1