INFINITE SERIES
317
37.43
If E a n and E b n are series of positive terms and
E b n not converge.
show by example that E a n may converge and
Let a n = l/n
2 and b n = \ln.
But, E 1/n
2 converges and E \ln diverges.
37.44
Determine whether
converges.
Use the limit comparison test with the convergent p-series
Then
Therefore, the given series is convergent.
37.45
Determine whether
is convergent.
Use the comparison test:
The geometric series
is convergent. Hence, the given
series is convergent.
is convergent.
Determine whether
37.46
Intuitively, we ignore the 1 in the denominator, so we use the limit comparison test with the divergent series
Hence, the given series is divergent.
37.47
Determine whether
is convergent.
Use the limit comparison test with the convergent p-series
since, by L'Hopital's rule,
Therefore, the given series is convergent.
37.48 Determine whether
is convergent.
for
Hence, the series converges, by comparison with the convergent geometric
series
37.49
Determine whether
converges.
Use the integral test with
Note that f(x) is decreasing, since
diverges.
Therefore, the given series
37.50
Determine whether
converges.
Use the integral test with
Hence, the series converges.
37.51
Give an example of a series that is conditionally convergent (that is, convergent but not absolutely convergent).
is convergent by the alternating series test (the terms are alternately
positive and negative, and their magnitudes decrease to zero). But
diverges.
for
f(x)=1/x(Inx)2.