All GRE Subject Test: Math Resources
Example Questions
Example Question #1 : Trigonometric Integrals
Evaluate:
1) The 1/2 is a constant, and so is pulled out front.
2) The integral of cos(x) is sin(x), by definition.
3) Writing the limits for evaluation:
4) Using the unit circle, , and .
5)Simplifying:
Example Question #3 : Concepts Of Convergence And Divergence
There are 2 series, and , and they are both convergent. Is convergent, divergent, or inconclusive?
Divergent
Inconclusive
Convergent
Convergent
Infinite series can be added and subtracted with each other.
Since the 2 series are convergent, the sum of the convergent infinite series is also convergent.
Note: The starting value, in this case n=1, must be the same before adding infinite series together.
Example Question #3 : Concepts Of Convergence And Divergence
You have a divergent series , and you multiply it by a constant 10. Is the new series convergent or divergent?
Inconclusive
Divergent
Convergent
Divergent
This is a fundamental property of series.
For any constant c, if is convergent then is convergent, and if is divergent, is divergent.
is divergent in the question, and the constant c is 10 in this case, so is also divergent.
Example Question #161 : Calculus
There are 2 series and .
Is the sum of these 2 infinite series convergent, divergent, or inconclusive?
Convergent
Divergent
Inconclusive
Convergent
A way to find out if the sum of the 2 infinite series is convergent or not is to find out whether the individual infinite series are convergent or not.
Test the first series
.
This is a geometric series with .
By the geometric test, this series is convergent.
Test the second series
.
This is a geometric series with .
By the geometric test, this series is convergent.
Since both of the series are convergent, is also convergent.
Example Question #162 : Calculus
Find the radius of convergence for the power series
We can use the limit
to find the radius of convergence. We have
This means the radius of convergence is .
Example Question #1 : Sequences
Determine if the following series is divergent, convergent or neither.
Inconclusive
Both
Divergent
Convergent
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 #1 : Sequences
Calculate the sum of the following infinite geometric series:
This is an infinite geometric series.
The sum of an infinite geometric series can be calculated with the following formula,
, where is the first value of the summation, and r is the common ratio.
Solution:
Value of can be found by setting
r is the value contained in the exponent
Example Question #3021 : Calculus Ii
Determine how many terms need to be added to approximate the following series within :
This is an alternating series test.
In order to find the terms necessary to approximate the series within first see if the series is convergent using the alternating series test. If the series converges, find n such that
Step 1:
An alternating series can be identified because terms in the series will “alternate” between + and –, because of
Note: Alternating Series Test can only show convergence. It cannot show divergence.
If the following 2 tests are true, the alternating series converges.
- {} is a decreasing sequence, or in other words
Solution:
1.
2. {} is a decreasing functon, since a factorial never decreases.
Since the 2 tests pass, this series is convergent.
Step 2:
Plug in n values until
4 needs to be added to approximate the sum within .
Example Question #2 : Limits Of Sequences
Evaluate: . (Round to 4 places)
Step 1: Plug in values into the function and add up the fraction:
Step 2: Find the sum of the fractions....
We can convert the fractions to decimals:
Step 3: Round to places...
Example Question #4 : Sequences
Which of the following are not infinite sequences?
Step 1: Define what an infinite sequence is...
An infinite sequence is a sequence that is non-terminating.
Step 2: Determine if each sequence above is infinite...
For , the sequence is always infinite because the set of factorials is infinite. Also, the set of values by raising two factorial powers together is also infinite, it never has an ending term.
For , this sequence is FINITE!
For , this sequence is FINITE!
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