All AP Calculus BC Resources
Example Questions
Example Question #11 : Series Of Constants
True or False, a -series cannot be tested conclusively using the ratio test.
False
True
True
We cannot test for convergence of a -series using the ratio test. Observe,
For the series ,
.
Since this limit is regardless of the value for , the ratio test is inconclusive.
Example Question #1 : Ratio Test And Comparing Series
Determine if the following series is divergent, convergent or neither.
Divergent
Inconclusive
Neither
Convergent
Both
Convergent
In 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 #2 : Ratio Test And Comparing Series
Determine if the following series is divergent, convergent or neither.
Divergent
Convergent
Inconclusive
Both
Neither
Divergent
In 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 #3 : Ratio Test And Comparing Series
Determine if the following series is divergent, convergent or neither.
Both
Neither
Inconclusive
Convergent
Divergent
Divergent
In 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 #4 : Ratio Test And Comparing Series
Determine if the following series is convergent, divergent or neither.
More tests are needed.
Convergent
Inconclusive
Divergent
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,
.
If
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.
Divergent
Inconclusive
Neither
More tests are needed.
Convegent
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,
.
If
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 #71 : Convergence And Divergence
Determine if the following series is convergent, divergent or neither.
Inconclusive
More tests needed.
Convergent
Divergent
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,
.
If
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 #7 : Ratio Test And Comparing Series
Determine if the following series is divergent, convergent or neither.
Inconclusive
Convergent
More tests are needed.
Divergent
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,
.
If
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.
Inconclusive.
More tests are needed.
Neither
Divergent
Convergent
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,
.
If
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 convergent.
, and divergent.
, and convergent.
, and neither.
, 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,
.
If
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.
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