Power series

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Last updated 2:16 AM on 6/12/26
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12 Terms

1
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What is a power series

A series of the form cn(x-a)n

<p>A series of the form c<sub>n</sub>(x-a)<sup>n</sup></p>
2
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<p>What are the 2 new variables in the power series</p>

What are the 2 new variables in the power series

Cn is a constant called the coefficient, a a constant called the center

3
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<p>solve this question</p>

solve this question

Remember that we need the absolute values, because they affect the answer

<p>Remember that we need the absolute values, because they affect the answer</p>
4
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How can you notice when your looking at a power series and not just something to the power of n

There has to be an x in the function

5
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What are the 3 types of ways a power series can converge

  • at 1 point

  • at all points

  • at an interval

6
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What is the interval of convergence

The set all values of x that converge

7
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What is the radius of convergence

It is the distance from the middle of the interval of convergence to one of the endpoints. (like half of the interval of convergence)

8
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What do you need to remember about where to keep the absolute value signs

Always keep the absolute value signs around the x term

9
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When finding the radius of convergence, what must you remember

The |x+something| stays on the left side of the greater/less than side and then you subtract the “something” from both sides of the greater/less than side to find the ends of the convergence.

<p>The |x+something| stays on the left side of the greater/less than side and then you subtract the “something” from both sides of the greater/less than side to find the ends of the convergence.</p>
10
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What is the alternating harmonic series

(-1)n/n

<p>(-1)<sup>n</sup>/n</p>
11
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What must you do at the end of finding the interval of convergence

You have to sub in the endpoints of the interval into the equation and see it converges.

12
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When does the ratio test converge or diverge

Converges when the function is less than 1

Diverges when the function is greater than 1