All AP Calculus BC Resources
Example Questions
Example Question #1 :Ratio Test And Comparing Series
Determine if the following series is divergent, convergent or neither.
Neither
我nconclusive
Convergent
Divergent
Both
Convergent
我n order to figure out if
is divergent, convergent or neither, we need to use the ratio test.
Remember that the ratio test is as follows.
Suppose we have a series. We define,
Then if
, the series is absolutely convergent.
, the series is divergent.
, the series may be divergent, conditionally convergent, or absolutely convergent.
Now lets apply the ratio test to our problem.
Let
and
Now
Now lets simplify this expression to
.
Since
.
We have sufficient evidence to conclude that the series is convergent.
Example Question #1 :Ratio Test And Comparing Series
Determine if the following series is divergent, convergent or neither.
Neither
Both
我nconclusive
Divergent
Convergent
Divergent
我n order to figure if
is convergent, divergent or neither, we need to use the ratio test.
Remember that the ratio test is as follows.
Suppose we have a series. We define,
Then if
, the series is absolutely convergent.
, the series is divergent.
, the series may be divergent, conditionally convergent, or absolutely convergent.
Now lets apply the ratio test to our problem.
Let
and
Now
.
Now lets simplify this expression to
.
Since,
we have sufficient evidence to conclude that the series is divergent.
Example Question #2 :Ratio Test And Comparing Series
Determine if the following series is divergent, convergent or neither.
Divergent
Convergent
Both
Neither
我nconclusive
Divergent
我n order to figure if
is convergent, divergent or neither, we need to use the ratio test.
Remember that the ratio test is as follows.
Suppose we have a series. We define,
Then if
, the series is absolutely convergent.
, the series is divergent.
, the series may be divergent, conditionally convergent, or absolutely convergent.
Now lets apply the ratio test to our problem.
Let
and
.
Now
.
Now lets simplify this expression to
.
Since,
we have sufficient evidence to conclude that the series is divergent.
Example Question #2 :Ratio Test And Comparing Series
Determine if the following series is convergent, divergent or neither.
Convergent
Divergent
我nconclusive
More tests are needed.
Neither
Divergent
To determine if
is convergent, divergent or neither, we need to use the ratio test.
The ratio test is as follows.
Suppose we a series. Then we define,
.
我f
the series is absolutely convergent (and therefore convergent).
the series is divergent.
the series may be divergent, conditionally convergent, or absolutely convergent.
Now lets apply this to our situtation.
Let
and
Now
We can rearrange the expression to be
Now lets simplify this.
When we evaluate the limit, we get.
.
Since, we have sufficient evidence to conclude that the series diverges.
Example Question #5 :Ratio Test And Comparing Series
Determine if the following series is divergent, convergent or neither.
我nconclusive
Divergent
Convegent
Neither
More tests are needed.
Convegent
To determine if
is convergent, divergent or neither, we need to use the ratio test.
The ratio test is as follows.
Suppose we a series. Then we define,
.
我f
the series is absolutely convergent (and thus convergent).
the series is divergent.
the series may be divergent, conditionally convergent, or absolutely convergent.
Now lets apply this to our situtation.
Let
and
Now
We can rearrange the expression to be
.
Now lets simplify this.
When we evaluate the limit, we get.
.
Since, we have sufficient evidence to conclude that the series converges.
Example Question #6 :Ratio Test And Comparing Series
Determine if the following series is convergent, divergent or neither.
我nconclusive
Neither
More tests needed.
Divergent
Convergent
Divergent
To determine if
is convergent, divergent or neither, we need to use the ratio test.
The ratio test is as follows.
Suppose we a series. Then we define,
.
我f
the series is absolutely convergent (therefore convergent).
the series is divergent.
the series may be divergent, conditionally convergent, or absolutely convergent.
Now lets apply this to our situtation.
Let
and
Now
We can rearrange the expression to be
Now lets simplify this.
When we evaluate the limit, we get.
.
Since, we have sufficient evidence to conclude that the series diverges.
Example Question #2 :Ratio Test And Comparing Series
Determine if the following series is divergent, convergent or neither.
More tests are needed.
Divergent
我nconclusive
Neither
Convergent
Divergent
To determine if
is convergent, divergent or neither, we need to use the ratio test.
The ratio test is as follows.
Suppose we a series. Then we define,
.
我f
the series is absolutely convergent (and thus convergent).
the series is divergent.
the series may be divergent, conditionally convergent, or absolutely convergent.
Now lets apply this to our situtation.
Let
and
Now
We can simplify the expression to be
When we evaluate the limit, we get.
.
Since, we have sufficient evidence to conclude that the series diverges.
Example Question #8 :Ratio Test And Comparing Series
Determine of the following series is convergent, divergent or neither.
Divergent
More tests are needed.
我nconclusive.
Convergent
Neither
Divergent
To determine whether this series is convergent, divergent or neither
we need to remember the ratio test.
The ratio test is as follows.
Suppose we a series. Then we define,
.
我f
the series is absolutely convergent (and therefore convergent).
the series is divergent.
the series may be divergent, conditionally convergent, or absolutely convergent.
Now lets apply this to our situtation.
Let
and
Now
We can rearrange the expression to be
Now lets simplify this to.
When we evaluate the limit, we get.
.
Since, we have sufficient evidence to conclude that the series is divergent.
Example Question #9 :Ratio Test And Comparing Series
Determine what the following series converges to using the ratio test and whether the series is convergent, divergent or neither.
, and neither.
, and divergent.
, and convergent.
, and neither.
, and convergent.
, and convergent.
To determine whether this series is convergent, divergent or neither
we need to remember the ratio test.
The ratio test is as follows.
Suppose we a series. Then we define,
.
我f
the series is absolutely convergent (thus convergent).
the series is divergent.
the series may be divergent, conditionally convergent, or absolutely convergent.
Now lets apply this to our situtation.
Let
and
Now
We can rearrange the expression to be
Now lets simplify this to.
When we evaluate the limit, we get.
.
Since, we have sufficient evidence to conclude that the series is convergent.
Example Question #1 :Ratio Test And Comparing Series
Determine the convergence or divergence of the following series:
The series is conditionally convergent.
The series is divergent.
The series (absolutely) convergent.
The series may be divergent, conditionally convergent, or absolutely convergent.
The series (absolutely) convergent.
To determine the convergence or divergence of this series, we use the Ratio Test:
我f, then the series is absolutely convergent (convergent)
我f, then the series is divergent
我f, the series may be divergent, conditionally convergent, or absolutely convergent
So, we evaluate the limit according to the formula above:
which simplified becomes
Further simplification results in
Therefore, the series is absolutely convergent.