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Lecture 29 – Power Series Def: The power series centered at x = a:
x is the variable and the c’s are constants (coefficients)
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For any power series, exactly one of the following is true:
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Example 1 – Radius and Interval of Convergence
Ratio Test: Series converges for
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Example 2 – Radius and Interval of Convergence
Ratio Test: Series converges for
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Example 3 – Radius and Interval of Convergence
Ratio Test:
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Example 3 – continued (testing endpoints)
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Example 4 – Radius and Interval of Convergence
Root Test:
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Example 4 – continued (testing endpoints)
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Example 5 – Radius and Interval of Convergence
Geometric Series:
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Example 5 – continued – what is the converging value?
Geometric Series:
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Lecture 30 – More Power Series
The geometric series: As a power series with a = 1, r = x and cn = 1 for all n: In other words, the function f(x) can be written as a power series. with
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Example 1 Create new power series for other functions through: and
sum, difference, multiplication, division, composition and differentiation and integration Example 1 with
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Example 2 with
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Consider the graphs:
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Example 3 with
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Need to solve for C. Set x = 0 to get:
Test endpoints???
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Example 4 with
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Need to solve for C. Set x = 0 to get:
Test endpoints???
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