how to represent certain types of functions as a power series

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Presentation transcript:

how to represent certain types of functions as a power series MACLAURIN SERIES how to represent certain types of functions as a power series You might wonder why we would ever want to express a known function as a sum of infinitely many terms. Integration. (Easy to integrate polynomials) Finding limit Finding a sum of a series (not only geometric, telescoping) How does the calculator find values of sine (or cosine or tangent)?

Maclaurin series ( center is 0 ) Example: Maclaurin series ( center is 0 ) Example: Find Maclaurin series

MACLAURIN SERIES Example: Find Maclaurin series Example: Find Maclaurin series Example: Find Maclaurin series Example: Find Maclaurin series Example: Find Maclaurin series

MACLAURIN SERIES Important Maclaurin Series and Their Radii of Convergence MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620 How to memorize them

Important Maclaurin Series and Their Radii of Convergence MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620 Denominator is n! even, odd Denominator is n odd

Maclaurin series ( center is 0 ) How to find a Maclaurin Series of a function Use the formula Use the known functions 1) Replace each x 2) Diff 3) integrate 3) Find a product between two

MACLAURIN SERIES TERM-081

MACLAURIN SERIES TERM-091

MACLAURIN SERIES TERM-101

MACLAURIN SERIES TERM-082

MACLAURIN SERIES TERM-102

MACLAURIN SERIES TERM-091

Find the sum of a series using Maclaurin Series Geometric + Telescoping

TAYLOR AND MACLAURIN Example: Find the sum of the series

MACLAURIN SERIES TERM-102

MACLAURIN SERIES TERM-082

MACLAURIN SERIES TERM-141

MACLAURIN SERIES TERM-131

MACLAURIN SERIES TERM-132

MACLAURIN SERIES Example: Find the sum Leibniz’s formula:

Important Maclaurin Series and Their Radii of Convergence MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620 Denominator is n! even, odd Denominator is n odd MAC

MACLAURIN SERIES Example: Find Maclaurin series

Integration using Maclaurin Series

MACLAURIN SERIES TERM-122

MACLAURIN SERIES TERM-141

Important Maclaurin Series and Their Radii of Convergence MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620 Denominator is n! even, odd Denominator is n odd