Page 273 - Calculus Workbook For Dummies

257
Chapter 13: Infinite Series: Welcome to the Outer Limits

3. Use the direct comparison test. It’s easy to show that the terms of the series in Step 2 are
greater than or equal to the terms of the divergent p-series, so it, and thus your given series,
diverges as well.
j ! 1 diverges.
3
=
n 1 n + n + lnn
3 1
Try the limit comparison test: Use the divergent harmonic series ! , as your benchmark.
n
=
n 1
1
n + n + lnn
lim
n " 3 1
n
n
= lim
n + n + lnn
n " 3
1
= lim (By L’Hôpital’s Rule)
n " 3 1 1
1 + + n
2 n
= 1
Because the limit is finite and positive, the limit comparison test tells you that ! 1
3
n 1 n + n + lnn
=
diverges with the benchmark series. By the way, you could do this problem with the direct
comparison test as well. Do you see how? Hint: You can use the harmonic series as your bench-
mark, but you have to tweak it first.
*k ! 1 3 converges.
3
3
n 1 n - ^ ln nh
=
1. Do a quick check to see whether the direct comparison test will give you an immediate
answer.
It doesn’t because ! 1 3 is larger than the known convergent p-series ! 1 3 .
3
3
3
=
=
n 1 n - ^ lnnh n 1 n
3
2. Try the limit comparison test with ! 1 3 as your benchmark.
n 1 n
=
1
3 3
n - ^ lnnh
lim
n " 3 1
n 3
n 3
= lim 3
n " 3 3
n - ^ lnnh
1
= lim 3
n " 3 ^ lnnh
1 - 3
n
1
= lim 3
n " 3 lnn
1 - c m
n
1
= 3 (Just take my word for it.)
lnn
1 - limc n m
n " 3
1 (Just take my word for it.)
= 3
lnn
1 - c lim n m
n " 3
1 (L’Hôpital’s Rule from Chapter 12)
=
J 1 N 3
K n O
1 - K K lim 1 O
O
n " 3
L P
= 1
3
Because this is finite and positive, the limit comparison test tells you that ! 1
3 lnnh 3
n - ^
=
converges with the benchmark series. n 1