# How do you find the Maclaurin Series for #sin^3 (x)#?

#= x^3-x^5/2+(13x^7)/120-(41x^9)/3024+...#

So:

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To find the Maclaurin series for ( \sin^3(x) ), we'll start with the Maclaurin series for ( \sin(x) ), which is ( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots ). Then, we'll cube this series term by term, and simplify to find the Maclaurin series for ( \sin^3(x) ).

( \sin^3(x) = (\sin(x))^3 )

Using the binomial theorem, we expand ( (\sin(x))^3 ) as ( \sin^3(x) = \sin(x) \cdot \sin^2(x) ).

Then, ( \sin^2(x) = (1 - \cos(2x))/2 ).

Finally, replace ( \sin(x) ) with its Maclaurin series and simplify to find the Maclaurin series for ( \sin^3(x) ).

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To find the Maclaurin series for ( \sin^3(x) ), we first need to express it in terms of a known Maclaurin series. We can use the trigonometric identity ( \sin^3(x) = (\sin(x))^3 ) and then expand ( \sin(x) ) using its Maclaurin series.

The Maclaurin series for ( \sin(x) ) is:

[ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots ]

To find ( (\sin(x))^3 ), we raise each term in the series to the power of 3 and then multiply the resulting series term by term.

[ (\sin(x))^3 = (x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots)^3 ]

Expanding this expression, we get the Maclaurin series for ( \sin^3(x) ).

Note that the Maclaurin series for ( \sin^3(x) ) may involve a large number of terms, as it requires expanding the cube of each term in the Maclaurin series for ( \sin(x) ) and then combining like terms.

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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.

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.

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.

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