All Flashcards
Define absolute convergence.
A series is absolutely convergent if converges.
Define conditional convergence.
A series is conditionally convergent if converges but diverges.
What does convergence mean?
A series converges if the sequence of its partial sums approaches a finite limit.
What does divergence mean?
A series diverges if the sequence of its partial sums does not approach a finite limit.
Define harmonic series.
A harmonic series is a series of the form .
What is an alternating series?
A series where the terms alternate in sign, often involving or .
Define p-series.
A p-series is a series of the form , where p is a constant.
What is the Direct Comparison Test?
A test to determine convergence or divergence by comparing a given series to a known convergent or divergent series.
What is the Alternating Series Test?
A test used to prove convergence of an alternating series if the absolute value of the terms decreases monotonically to zero.
Define sequence.
An ordered list of numbers.
Explain the first step in determining absolute or conditional convergence.
First, take the absolute value of the terms in the series, i.e., consider .
What does it mean if converges?
If converges, then is absolutely convergent.
What does it mean if diverges, but converges?
If diverges, but converges, then is conditionally convergent.
Why test for absolute convergence first?
It's often easier to determine absolute convergence first. If a series is absolutely convergent, you don't need to check for conditional convergence.
How does the Alternating Series Test relate to conditional convergence?
The Alternating Series Test can be used to show that an alternating series converges. If the absolute value of that series diverges, then the original series is conditionally convergent.
Explain the role of comparison tests in determining absolute convergence.
Comparison tests, like the Direct Comparison Test, can be used to determine if converges or diverges, thus helping to establish absolute convergence.
How does the behavior of affect convergence?
Since , taking the absolute value means . This is useful for comparison tests.
What is a harmonic series, and why is it important in convergence tests?
A harmonic series is , and it's a classic example of a divergent series. It's often used for comparison.
Explain the difference between absolute and conditional convergence in terms of error estimation.
Absolutely convergent series have better error estimation properties than conditionally convergent series, as rearranging terms in a conditionally convergent series can change its sum.
How do you handle series that are not alternating but also not strictly positive?
Take the absolute value of the terms and then apply convergence tests. If the absolute value converges, the series is absolutely convergent.
How to determine if is absolutely or conditionally convergent?
- Take absolute value: . 2. This is a convergent p-series (p=2). 3. Therefore, the series is absolutely convergent.
How to determine the convergence of ?
- Take the absolute value: . 2. Since , compare to . 3. The p-series converges, so the original series is absolutely convergent.
Steps to check for conditional convergence.
- Verify the series converges using Alternating Series Test. 2. Take the absolute value of the terms. 3. Show the absolute value series diverges. 4. Conclude it's conditionally convergent.
How to test for absolute/conditional convergence?
- Alternating Series Test shows convergence. 2. Absolute value gives , a divergent p-series (p=1/2). 3. Conditionally convergent.
How to test for absolute/conditional convergence?
- Alternating Series Test shows convergence. 2. Absolute value gives , which diverges by Limit Comparison Test with . 3. Conditionally convergent.
How to test for absolute/conditional convergence?
- Take absolute value: . 2. Since , compare to . 3. Ratio Test shows converges. 4. Absolutely convergent.
How to test for absolute/conditional convergence?
- Alternating Series Test shows convergence. 2. Absolute value gives , which diverges by Comparison Test with . 3. Conditionally convergent.
How to test for absolute/conditional convergence?
- Take absolute value: . 2. Apply Ratio Test. 3. The series converges absolutely.
How to test for absolute/conditional convergence?
- Alternating Series Test shows convergence. 2. Absolute value gives , which converges by Limit Comparison Test with . 3. Absolutely convergent.
How to test for absolute/conditional convergence?
- Alternating Series Test fails since , so the series diverges. 2. No need to check absolute convergence.