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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
Definition
|
|
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
|
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] |
|
|
