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| Conceptually, what is a power series? |
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Definition
| A power series is a polynomial with an infinite number of terms. |
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| Conceptually, what is a the center of a power series? |
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Definition
| The center of a power series is the horizontal shift of the series. |
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| Conceptually, what does it mean for a power series to converge to a function? |
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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. |
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| What is an interval of convergence? |
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Definition
| The interval of convergence is the values of x for which a series converges to its generating function |
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| What are the three scenarios under which a power series can converge to a function? |
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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. |
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| When you must check end points when finding the interval of convergence? |
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Definition
| You must check endpoints whenever the series is non-geometric. |
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| How do you find an interval of convergence? |
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Definition
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 |
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| What is an nth order Taylor Polynomial generated by f(x) at x=a? |
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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. |
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| What is a Taylor Polynomial used for? |
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Definition
| A Taylor Polynomial is used to approximate a function. It's like a tangent line on steroids. |
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| What is the difference between a Taylor Polynomial and a Taylor Series |
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Definition
| A Taylor Polynomial has a finite number of terms. A Taylor Series has an infinite number of terms. |
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Definition
| A Taylor Series is a power series that converges to a function on some interval of x. |
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| What information do you get from a Taylor Series? |
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Definition
| You get the y-coordinate and an infinite number of derivatives of f(x) at x=a. |
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What is the nth derivative (at the center) of
[image]? |
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Definition
| The nth derivative of this series is 1. |
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| What is a Maclaurin Series? |
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Definition
| A Maclaurin Series is a Taylor Series centered at 0. |
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| How do you use a Taylor Polynomial to approximate a function? |
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Definition
| Find the equation of the desired Taylor Polynomial, then substitute the value of x. |
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| What is the relationship between a tangent line at x=a and a Taylor Series at x=a? |
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Definition
| The tangent line is equivalent to the first order Taylor Polynomial. |
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| What are the six methods for creating a power series that converges to a function? |
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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. |
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