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Questions and Answers
What is a power series?
What is a power series?
- A series where each term is a polynomial (correct)
- A series where each term is a trigonometric function
- A series where each term is a constant
- A series where each term is an exponential function
When does a power series converge?
When does a power series converge?
- When the terms alternate in sign
- When the terms form a geometric sequence
- When the terms satisfy the ratio test or root test (correct)
- When the terms approach zero
What is the radius of convergence of a power series?
What is the radius of convergence of a power series?
- The point where the series alternating in sign
- The interval where the series diverges
- The distance from the origin to the closest singularity (correct)
- The sum of the absolute values of the coefficients
What is the general form of a power series?
What is the general form of a power series?
What happens if a power series converges at one point outside the interval of convergence?
What happens if a power series converges at one point outside the interval of convergence?
What is the significance of the radius of convergence in a power series?
What is the significance of the radius of convergence in a power series?
Which of the following is the general form of a power series?
Which of the following is the general form of a power series?
What is the radius of convergence for the power series $rac{x^n}{n!}$?
What is the radius of convergence for the power series $rac{x^n}{n!}$?
If a power series converges at $x=3$, what can be concluded about its interval of convergence?
If a power series converges at $x=3$, what can be concluded about its interval of convergence?
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Study Notes
Power Series
- A power series is a type of infinite series of the form ∑[a_n(x-a)^n] where 'a' is a constant, 'x' is the variable, and 'a_n' are constants.
Convergence of a Power Series
- A power series converges when the sum of the terms approaches a finite limit as the number of terms increases.
- The convergence of a power series depends on the value of 'x' and the constants 'a_n'.
Radius of Convergence
- The radius of convergence is the distance from the center of convergence to the nearest point where the series diverges.
- It is a measure of how far the power series converges from the center point.
General Form of a Power Series
- The general form of a power series is ∑[a_n(x-a)^n] where 'a' is a constant, 'x' is the variable, and 'a_n' are constants.
Convergence Outside the Interval of Convergence
- If a power series converges at one point outside the interval of convergence, the series converges absolutely for all points within the circle of convergence.
Significance of Radius of Convergence
- The radius of convergence determines the range of values of 'x' for which the power series converges.
- It is essential to determine the radius of convergence to know the region where the power series is valid.
Radius of Convergence for a Specific Power Series
- The radius of convergence for the power series ∑[x^n/n!] is infinite, meaning it converges for all values of 'x'.
Convergence and Interval of Convergence
- If a power series converges at x=3, then the series converges absolutely for all points within the circle of convergence centered at x=3, and the radius of convergence is at least 3.
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