Page 276 - Calculus Workbook For Dummies

260 Part IV: Integration and Infinite Series

3
*q ! ! n n converges.
n 1 n
=
There’s a factorial, so try the ratio test:
n
^ n + 1h ! n
y = limc m
+
n 1 n " 3 n + 1
^ n + 1h
lim n
n " 3 ! n lny = ln limc n m o
e
n n n " 3 n + 1
^ n + 1h ! n n n
= lim n
+
e
n " 3 ! n n + 1h n 1 = lim ln c n + 1 m o
^
n " 3
^ n + 1h n n J N
= lim n 1 K ln c n m O
n " 3 + n + 1
^ n + 1h limK O
=
O
n n n " 3 K K 1 O
= lim n n
n " 3 ^ n + 1h L P
J N
1
2 O
n K n + 1 $ n + - n
n n
= limc m K ^ n + 1h O
n " 3 n + 1 = lim (L’Hôpital’s Rule)
n " 3 K - 1 O
K K n 2 O O
Finish in the right L P
column with = limc - n m
n " 3 n + 1
logarithmic
differentiation. lny = - 1
y = e - 1
Because this is less than 1,
the series converges.
3 3 n
*r ! c m converges.
n
n 1 4
=
3 3 n
/n
1
Rewrite this so it’s one big nth power: !c n $ m . Now look at the limit of the nth root.
n 1 4
=
1 /n
n
3 1 /n
lim c e n m o
n " 3 4
3 1
= lim n /n
n " 3 4
3 1 /n 0
= limn _ A ’ L Hopital ’s Rule case 3 i
:
4 n " 3
3 1 /n
lny = ln c limn m
4 n " 3
3 1 /n
= ln + lim lnn i
_
4 n " 3
3 lnn
= ln + lim
4 n " 3 n
1
3 n
= ln + lim _ ’ L Hopital ’s Rulei
4 n " 3 1
3
lny = ln
4
3
y =
4
3
Thus the limit of the nth root is and so the series converges.
4