Page 275 - Calculus Workbook For Dummies

259
Chapter 13: Infinite Series: Welcome to the Outer Limits

3
2. Try the following nifty trick. Ignore the first three terms of ! n 3 , which doesn’t affect
n 1 ! n 3 3 3
=
4 5 6
the convergence or divergence of the series. The series is now + + + ..., which can
3 ! 4 ! 5 ! 6
3
be written as ! ^ n + 3h .
n 1 ^ n + 3h !
=
3. Try the limit comparison test again.
3
^ n + 3h
^ n + 3h !
lim
n " 3 1
! n
3
! n n + 3h
^
= lim
n " 3 ^ n + 3h !
3
^ n + 3h
= lim
n " 3 ^ n + 3 ^h n + 2 ^h n + 1h
3
n + lesser powers of n
= lim 3
n " 3 n + lesser powers of n
= 1 _ by the horizontalasymptote rulei
3
3
3
Thus, ! ^ n + 3h converges by the limit comparison test. And because ! n 3 is the same
n 1 ^ n + 3h ! n 1 ! n
=
=
series except for its first three terms, it converges as well.
3
o ! 1 n converges.
=
n 1 ` ln n + 2hj
^
J N 1 /n
K 1 O
lim K n O
n " 3 O
^
K ` ln n + 2hj
L P
1
Try the root test: lim
=
^
n " 3 ln n + 2h
= 0
This is less than 1, so the series converges.
3
p ! n n n converges.
n 1 n
=
Try the root test again:
J N 1 /n
/n
lim K n n n O = lim n n n / 1 2 = lim / 1 2 - 1 n /n = 0
n " 3 K K
O O
n " 3
n " 3
L n P n
Thus the series converges.