Problem :
Determine if the series
converges.
If
a
_{n}
is the
n
th term of the series, then
For large
n
, this quotient is very close to
1/2
. Hence, for all
but the first few integers
n
, the quotient will be
≤3/4
.
Since we may disregard the first few terms when considering questions
of convergence, the ratio test implies that the series converges.
Problem :
Suppose
x
is a fixed real number. Determine whether or not the series
converges.
Letting
a
_{n}
equal the
n
th term of the series, we have
Since this ratio is clearly less than, say,
1/2
for all but the
first few
n
, the ratio test implies that it converges.
Problem :
Show that the series
1/r
^{n}


converges for any real number
r > 1
.
If
a
_{n}
is the
n
th term of the series, then
Since
1/r < 1
whenever
r > 1
, the ratio test implies that the
series converges.