Shared Flashcard Set

Details

Chapter 9 current set
Full set of Series cards
78
Mathematics
12th Grade
01/29/2022

Additional Mathematics Flashcards

 


 

Cards

Term
What is a series of constants?
Definition
A series of constants is the sum of an infinite number of numbers.
Term
By definition, when does a series converge/diverge?
Definition
A series converges when the sequence of its partial sums converges, and diverges when the sequence of its partial sums diverges.
Term
When do we use the definition to prove that a series converges or diverges?
Definition
After everything else has failed but before you've cried.
Term
Explain the conceptual difference between a convergent/divergent series and a convergent/divergent sequence.
Definition
A SEQUENCE converges when the terms approach a number, and diverges when the terms do not approach a number.

A series converges when the terms ADD UP to a number, and diverges when the terms to NOT add up to a number.
Term
What are the ways of proving that a series of constants converges/diverges?
Definition
1) GST
2) nth Term Test
3) p-series Test
4) Alternating Series Test
5) Direct Comparison Test
6) Limit Comparison Test
7) Integral Test
8) Absolute Convergence Test
9) Ratio Test
10) Definition
Term
What are the conditions for the GST?
Definition
The series must be geometric
Term
Explain how to use the Ratio Test to prove that a series of constants converges/diverges.
Definition
1) Take the limit of the ratio of the n+1st term to the nth term as n->infinity.
2) If that limit is greater than one, the series diverges
3) If that limit is less than one, the series converges.
4) If the ratio is 1, then try another test of convergence.
Term
What are the conditions to use the Ratio Test?
Definition
The series must be a series of positive terms.
Term
Explain how to use the nth Term Test to prove a series converges or diverges.
Definition
1) Take the limit of the general term.

2) If the limit is anything other than zero, the series diverges.

3) If the limit is zero, then the test is inconclusive.
Term
What does it mean for a series to converge absolutely?
Definition
A series converges absolutely if the absolute value of the series converges.
Term
What does it mean if a series converges conditionally?
Definition
A series converges conditionally if the series converges, but does NOT converge absolutely.
Term
What are the conditions for the P-Series Test?
Definition
The series must be a p-series of positive terms.
Term
How do you use the P-Series Test to prove that a series converges or diverges?
Definition
1) Establish that the conditions have been met.

2) If [image], the series converges.

3) If [image], the series diverges.
Term
When should you consider using the Absolute Convergence Test?
Definition
You should consider using the ACT when the series has negative terms.
Term
When should you consider using the nth Term Test?
Definition
You should use the nth Term Test when the terms are obviously not going to zero
Term
When should consider using the Ratio Test?
Definition
You should use the Ratio Test when the other tests fail or when there is an n! involved
Term
What are the conditions for the DCT?
Definition
The given series and your chosen series must both be series of non-negative terms.
Term
How do you use the DCT to prove that a series converges/diverges?
Definition
Establish that the conditions have been met.

If you suspect that the series converges, compare it to a larger convergent series.

If you suspect the series diverges, compare it to a smaller divergent series.
Term
What are the conditions for the LCT?
Definition
The given series and chosen series must both be series of positive terms.
Term
How do you use the LCT to prove that a series of constants converges or diverges?
Definition
Take the limit of the ratio of the general terms of the given series and the your chosen series.

1) If the faster series converges, then the slower series converges.

2) If the slower series diverges, then the faster series diverges as well.

3) If both series grow at the same rate, then both series either converge or diverge.
Term
How do you use the Alternating Series Test to show that a series converges/diverges?
Definition
Establish that the series is alternating, decreases in absolute value, and that the limit of the absolute value of the general term is zero. Conclude that the series converges.

You cannot use this test to show that a series diverges.
Term
How do you use the GST to prove a series converges or diverges?
Definition
1) State that the series is geometric.
2) Find the absolute value of the ratio.
3) If the absolute value of r is less than one, the series converges. Otherwise the series diverges.
Term
How do you use the ACT to prove a series converges or diverges?
Definition
1) Take the absolute value of the general term to produce a new series.
2) If that series converges, the original series converges absolutely and therefore converges.
3) If the new series diverges, the test is inconclusive.
Term
Geometric Series Test (GST)
Definition
If the series is geometric with constant ratio r, then

the series converges when [image],

and the series diverges when [image]
Term
Ratio Test
Definition
If [image] is a series of positive terms, then:

the series converges when [image],

the series diverges when [image], and

the test is inconclusive when [image].
Term
nth Term Test
Definition
[image] diverges if [image].

The test is inconclusive when [image].
Term
Absolute Convergence Test (ACT)
Definition
If [image] converges, then [image] converges.

If [image] diverges, then the test is inconclusive.
Term
P-Series Test
Definition
If [image] is a series of positive terms, and [image], where p is a constant, then:

the series converges when [image] and

the series diverges when [image]
Term
Direct Comparison Test (DCT)
Definition
Let [image] be a known series, and [image] and [image] be series of positive terms. Then:

[image] converges if [image] for all [image] and [image] converges.

[image] diverges if [image] and [image] diverges.

Otherwise the test is inconclusive.
Term
Limit Comparison Test (LCT)
Definition
Let [image] and [image] be series of positive terms.

If [image], where c is non-zero, then both series either converge or diverge.

If [image] and [image] converges, then [image] conerges as well.

If [image], and [image] diverges, then [image] diverges as well.
Term
How should you decide which test to use?
Definition
1) If the terms are obviously not going to zero, use the nth term test.

2) If the series is geometric, use the GST.

3) If the series is a p-series, then use the P-Series Test

4) If the series alternates, try AST

5) If none of the above work, consider the LCT, DCT, or Integral Test

6) THEN use the Ratio Test. Combine with ACT if necessary.

7) If absolutely nothing else works, use the definition.
Term
Conceptually, what is a power series?
Definition
A power series is a polynomial with an infinite number of terms.
Term
Conceptually, what is a the center of a power series?
Definition
The center of a power series is the horizontal shift of the series.
Term
Conceptually, what does it mean for a power series to converge to a function?
Definition
If a power series converges to a function (on an interval of convergence), then as you increase the number of terms, the graph of the polynomial gets closer to the graph of the function. And for every value of x in the IOC, the sum of the series (of constants) will be equivalent to the y-coordinate of the corresponding function at the same x.
Term
What is an interval of convergence?
Definition
The interval of convergence is the values of x for which a series converges to its generating function
Term
What are the three scenarios under which a power series can converge to a function?
Definition
A power series can converge to a function on a specific interval of x, for all values of x, or at the center only.
Term
When you must check end points when finding the interval of convergence?
Definition
You must check endpoints whenever the series is non-geometric.
Term
How do you find an interval of convergence?
Definition
If the series is geometric, then use GST. The interval will be open.

If you took the derivative or antiderivative of a series with a known interval of convergence, keep the same open interval, but check the endpoints.

1) Combine the ACT and Ratio Test to take the limit of the absolute value of the ratio (as n->infinity)

2) Set the limit to be <1 and solve to find the values of x for which the series converges.

3) Check the endpoints by using another Test of Convergence
Term
How do you use the ACT to prove a series converges or diverges?
Definition
1) Take the absolute value of the general term to produce a new series.
2) If that series converges, the original series converges absolutely and therefore converges.
3) If the new series diverges, the test is inconclusive.
Term
Geometric Series Test (GST)
Definition
If the series is geometric with constant ratio r, then

the series converges when [image],

and the series diverges when [image]
Term
Ratio Test
Definition
If [image] is a series of positive terms, then:

the series converges when [image],

the series diverges when [image], and

the test is inconclusive when [image].
Term
nth Term Test
Definition
[image] diverges if [image].

The test is inconclusive when [image].
Term
Absolute Convergence Test (ACT)
Definition
If [image] converges, then [image] converges.

If [image] diverges, then the test is inconclusive.
Term
P-Series Test
Definition
If [image] is a series of positive terms, and [image], where p is a constant, then:

the series converges when [image] and

the series diverges when [image]
Term
Direct Comparison Test (DCT)
Definition
Let [image] be a known series, and [image] and [image] be series of positive terms. Then:

[image] converges if [image] for all [image] and [image] converges.

[image] diverges if [image] and [image] diverges.

Otherwise the test is inconclusive.
Term
Limit Comparison Test (LCT)
Definition
Let [image] and [image] be series of positive terms.

If [image], where c is non-zero, then both series either converge or diverge.

If [image] and [image] converges, then [image] conerges as well.

If [image], and [image] diverges, then [image] diverges as well.
Term
Integral Test
Definition
Let [image] be a series of positive terms, where [image], such that [image] is positive, decreasing, and continuous for all [image] for some positive integer N, then

[image] and [image] either both converge or both diverge.
Term
Alternating Series Test (AST)
Definition
Let [image] be:
1) an alternating series
2) whose terms decrease in absolute value
3) such that [image].

Then [image] converges.

If any of the above three coniditions is false, the test is inconclusive.
Term
Use GST when...
Definition
the series is geometric.
Term
Use the Ratio Test when...
Definition
the series is non-geometric and nothing else works, or if there are factorials involved.
Term
Use the nth-Term Test when...
Definition
the terms of the series are obviously not going to zero.
Term
Use the Absolute Convergence Test when...
Definition
taking the absolute value of a series would create a convenient series of positive terms. This test is almost never used on its own. Rather it is often combined with another test (like Ratio Test)
Term
Use the P-series Test when...
Definition
the series is a p-series. Combine with ACT as needed.
Term
Use the Direct Comparison Test when...
Definition
the series is either obviously larger than a simple divergent series, or obviously smaller than a simpler convergent series.
Term
Use the Limit Comparison Test when...
Definition
the EBM of the series is easily understood.
Term
Use the Integral Test when...
Definition
you can solve the corresponding improper integral.
Term
Use the Alternating Series Test when...
Definition
the series is alternating and its terms are obviously getting closer to zero.
Term
How should you decide which test to use?
Definition
1) If the terms are obviously not going to zero, use the nth term test.

2) If the series is geometric, use the GST.

3) If the series is a p-series, then use the P-Series Test

4) If the series alternates, try AST

5) If none of the above work, consider the LCT, DCT, or Integral Test

6) THEN use the Ratio Test. Combine with ACT if necessary.

7) If absolutely nothing else works, use the definition.
Term
Conceptually, what is a power series?
Definition
A power series is a polynomial with an infinite number of terms.
Term
Conceptually, what is a the center of a power series?
Definition
The center of a power series is the horizontal shift of the series.
Term
Conceptually, what does it mean for a power series to converge to a function?
Definition
If a power series converges to a function (on an interval of convergence), then for every value of x in the IOC, the sum of the series (of constants) will be equivalent to the y-coordinate of the corresponding function at the same x.
Term
What is an interval of convergence?
Definition
The interval of convergence is the values of x for which a series converges to its generating function
Term
What are the three scenarios under which a power series can converge to a function?
Definition
A power series can converge to a function on a specific interval of x, for all values of x, or at the center only.
Term
When you must check end points when finding the interval of convergence?
Definition
You must check endpoints whenever the series is non-geometric.
Term
When should you consider using the Absolute Convergence Test?
Definition
You should consider using ACT when the series has negative terms.
Term
What is an nth order Taylor Polynomial generated by f(x) at x=a?
Definition
An nth order Taylor Polynomial generated by f(x) at x=a is a polynomial that shares the same y-coordinate and the same first n derivatives as f(x) at x=a.
Term
What is a Taylor Polynomial used for?
Definition
A Taylor Polynomial is used to approximate a function. It's like a tangent line on steroids.
Term
What is the difference between a Taylor Polynomial and a Taylor Series
Definition
A Taylor Polynomial has a finite number of terms. A Taylor Series has an infinite number of terms.
Term
What is a Taylor Series?
Definition
A Taylor Series is a power series that converges to a function on some interval of x.
Term
What information do you get from a Taylor Series?
Definition
You get the y-coordinate and an infinite number of derivatives of f(x) at x=a.
Term
What is the nth derivative (at the center) of
[image]?
Definition
The nth derivative of this series is 1.
Term
What is a Maclaurin Series?
Definition
A Maclaurin Series is a Taylor Series centered at 0.
Term
How do you use a Taylor Polynomial to approximate a function?
Definition
Find the equation of the desired Taylor Polynomial, then substitute the value of x.
Term
What is the relationship between a tangent line at x=a and a Taylor Series at x=a?
Definition
The tangent line is equivalent to the first order Taylor Polynomial.
Term
What are the six methods for creating a power series that converges to a function?
Definition
1) Use the sum of a geometric series.
2) Differentiate a convergent series.
3) Integrate a convergent series.
4) Substitute an expression of x for x.
5) Multiply (or divide) by an expression of x.
6) Use the Taylor Series general term to create a Taylor series from scratch.
Term
What is the truncation error (function) and what does it measure?
Definition
The truncation error function is the sum of the terms of a series AFTER the Taylor Polynomial. It measures the accuracy of a Taylor polynomial approximation at any given value of x. By Taylor's theorem, the truncation error is:

[image]
Term
What is magnitude of error, and what does it measure?
Definition
The magnitude of error is the absolute value of the truncation error. It measures the margin of error, or the accuracy of the truncation error without factoring in overestimates or underestimates.
Term
Conceptually, what is the reationship between a Taylor Series, [image] and [image]?
Definition
If the series converges to f(x), then:
[image]
Supporting users have an ad free experience!