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How do you find the Maclaurin Series for #e^(sinx)#?

Answer 1

Samantha Adkison

#e^sinx = e^(sin(0))+ x cos(0) e^(sin(0)) + 1/2 x^2(cos^2 (0) - cos(0) sin(0))e ^(sin(0)) +....=1 + x+ x^2/2 +.....#

by applying the formula f(x) = f(0) + xf'(0) + x^2/2 f''(0) + higher order terms, which are also similarly calculable.

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Answer 2

Luke Alvarez

To find the Maclaurin series for ( e^{\sin(x)} ), you would use the formula for the Maclaurin series of a composite function. Here's how you do it:

Begin with the Maclaurin series for ( e^x ), which is ( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} ).
Replace ( x ) with ( \sin(x) ) in the series, giving ( e^{\sin(x)} = \sum_{n=0}^{\infty} \frac{\sin^n(x)}{n!} ).
Expand ( \sin^n(x) ) using the binomial theorem or another suitable method.
Simplify the resulting expression to obtain the Maclaurin series for ( e^{\sin(x)} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.