INFINITE SERIES
37.78
Test
37.79 Test
for convergence.
series converges.
37.81
Test
37.76 Study the convergence of
321
This is the series
comparison test with
By the alternating series test, it is convergent. By the limit
is divergent,
diverges. Hence, the given series is conditionally convergent.
Since
37.77 Prove the following special case of the ratio test: A series of positive terms £ a n is convergent if
Choose r so that
There exists an integer k such that, if
then
and, therefore,
So, if
Hence,
Hence, by comparison with the convergent geometric series
the series
is convergent, and, therefore, the given series is convergent (since it is obtained
from a convergent sequence by addition of a finite number of terms).
Use the limit comparison test with the convergent p-series
Hence, the given series converges.
Use the ratio test.
In
Therefore, the series diverges.
since In2<
37.80 Test
for convergence.
Use the ratio test.
Therefore, the
Use the ratio test.
series diverges.
Therefore, the
37.82 Determine the nth term of and test for convergence the series
Use the limit comparison test with the convergent p-series T, 1 In
2 .
The nth term is
Therefore, the given series converges.
37.83 Determine the nth term of and test for convergence the series
The nth term is l/(n + l)(n+ 2)-• • (2n). The nth term is less than 1/2-2-•-2 = 1/2". Hence, the
series is convergent by comparison with the convergent geometric series
or convergence.
for convergence.