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L27 Taylor and MaClaurin I

# L27 Taylor and MaClaurin I - Lecture 27 Taylor and...

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Lecture 27: Taylor and Maclaurin Se- ries We’ve shown that when a power series converges, it de±nes a function in x . In the next two lec- tures, we will see that many familiar functions such as sin x; cos x;e x ±±± can be represented as power series. Key Questions: If f is a function that can be represented by a power series, how do you con- struct the power series representation? For instance, how do we construct a power series for the function f ( x ) = e x 1

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If f is a di±erentiable function that can be represented by a power series, f ( x ) = c 0 + c 1 ( x ± a ) + c 2 ( x ± a ) 2 + ²²² ; j x ± a j < R; then we have f ( a ) = c 0 f 0 ( x ) = c 1 + 2 c 2 ( x ± a ) + 3 c 3 ( x ± a ) 2 + ²²² f 0 ( a ) = c 1 f 00 ( x ) = 2 c 2 + 2 ² 3 c 3 ( x ± a ) + 3 ² 4 c 4 ( x ± a ) 2 + ²²² f 00 ( a ) = f 000 ( x ) = 2 ² 3 c 3 + 2 ² 3 ² 4 c 4 ( x ± a ) + ²²² f 000 ( a ) = f ( n ) ( a ) = In other words, c n = 2
Theorem: If f has a power series representa- tion, f ( x ) = 1 X n =0 c n ( x ± a ) n ; j x ± a j < R; then its derivatives f ( n ) ( x ) can be obtained by the term-by-term di±erentiation of the series, f ( n ) ( a ) = n ! c n .

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L27 Taylor and MaClaurin I - Lecture 27 Taylor and...

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