All Flashcards
Define a sequence.
A function whose domain is the set of natural numbers.
Define a series.
The sum of the terms of a sequence.
What is a convergent sequence/series?
Terms approach a specific value (the limit).
What is a divergent sequence/series?
Terms do not approach a specific value; the sum is infinite.
Define an arithmetic sequence.
Sequence with a constant difference between consecutive terms.
Define a geometric sequence.
Sequence where each term is the previous term multiplied by a constant ratio.
Define a harmonic series.
Series where terms are reciprocals of positive integers.
Define a power series.
Series of the form where are constants and is a constant.
Define an alternating series.
Series where the terms alternate in sign.
Define a Taylor series.
Representation of a function as an infinite sum of terms, with each term being a polynomial function of a single variable.
Define a Maclaurin series.
A Taylor series centered at 0.
What does the Alternating Series Test state?
If is decreasing and , then the alternating series converges.
What does the Ratio Test state?
If , then the series converges if , diverges if , and is inconclusive if .
What does the nth Term Test for Divergence state?
If , then the series diverges.
How to determine convergence/divergence using the nth Term Test?
- Find . 2. If the limit is not 0, the series diverges. 3. If the limit is 0, the test is inconclusive.
How to apply the Limit Comparison Test?
- Choose a series to compare with. 2. Find . 3. If is finite and positive, both series converge or diverge together.
How to apply the Direct Comparison Test?
- Find a series to compare. 2. Establish inequality. 3. If larger converges, smaller converges. If smaller diverges, larger diverges.
How to apply the Integral Test?
- Verify is continuous, positive, decreasing. 2. Evaluate . 3. If the integral converges, the series converges. If the integral diverges, the series diverges.
How to apply the Alternating Series Test?
- Check if terms decrease in absolute value. 2. Check if . 3. If both conditions are met, the series converges.
How to apply the Ratio Test?
- Find . 2. If , the series converges. 3. If , the series diverges. 4. If , the test is inconclusive.
How to find the radius of convergence of a power series?
- Use the Ratio Test. 2. Solve for . 3. is the radius of convergence.
How to find the interval of convergence of a power series?
- Find the radius of convergence, R. 2. Test the endpoints and for convergence. 3. Write the interval, including or excluding endpoints based on convergence.
How to estimate the sum of an alternating series with a specified error?
- Use Alternating Series Error Bound: . 2. Find the smallest such that is less than the specified error. 3. Sum the first terms.
How to find a Taylor series for a given function?
- Find derivatives of the function. 2. Evaluate derivatives at the center. 3. Plug into the Taylor series formula: .