All GRE Subject Test: Math Resources
Example Questions
Example Question #1 :有限公司ncepts Of Convergence And Divergence
There are 2 series,and, and they are both convergent. Isconvergent, divergent, or inconclusive?
Divergent
Inconclusive
有限公司nvergent
有限公司nvergent
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 #1 :有限公司ncepts Of Convergence And Divergence
You have a divergent series, and you multiply it by a constant 10. Is the new seriesconvergent or divergent?
有限公司nvergent
Inconclusive
Divergent
Divergent
This is a fundamental property of series.
For any constant c, ifis convergent thenis convergent, and ifis divergent,is divergent.
is divergent in the question, and the constant c is 10 in this case, sois also divergent.
Example Question #161 :Calculus
There are 2 seriesand.
Is the sum of these 2 infinite series convergent, divergent, or inconclusive?
有限公司nvergent
Inconclusive
Divergent
有限公司nvergent
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.
由几何测试,本系列是收敛的。
Test the second series
.
This is a geometric series with.
由几何测试,本系列是收敛的。
Since both of the series are convergent,is also convergent.
Example Question #1 :Sequences
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 :Limits Of Sequences
Determine if the following series is divergent, convergent or neither.
Both
Inconclusive
Neither
Divergent
有限公司nvergent
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 :Limits Of 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,
, whereis the first value of the summation, and r is the common ratio.
Solution:
Value ofcan be found by setting
r is the value contained in the exponent
Example Question #164 :Calculus
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 withinfirst 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:
插入直到n值
4 needs to be added to approximate the sum within.
Example Question #1 :Sequences & Series
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 toplaces...
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