How do you find the Maclaurin Series for #sin(x^2)/x#?
In order to find the Maclaurin series of
We know that the Maclaurin series of
Given this known Maclaurin series, we can now see that
Note that we still have another factor we have to consider, namely the division by
Since we now have the series
All we need to do is to divide by an
We can check the first
Graph of
graph{sin(x^2)/x [10, 10, 5, 5]}
Graph of
graph{xx^(5)/(3!)+x^(9)/(5!)x^(13)/(7!) [10, 10, 5, 5]}
Overlapping both graphs produces the following:
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To find the Maclaurin series for ( \frac{\sin(x^2)}{x} ), follow these steps:
 Find the Maclaurin series for ( \sin(x^2) ).
 Divide each term of the resulting series by ( x ).
 Simplify the series to obtain the Maclaurin series for ( \frac{\sin(x^2)}{x} ).
Let's go through these steps:
 The Maclaurin series for ( \sin(x) ) is: [ \sin(x) = x  \frac{x^3}{3!} + \frac{x^5}{5!}  \frac{x^7}{7!} + \cdots ]
Substitute ( x^2 ) for ( x ) to get the Maclaurin series for ( \sin(x^2) ): [ \sin(x^2) = x^2  \frac{x^6}{3!} + \frac{x^{10}}{5!}  \frac{x^{14}}{7!} + \cdots ]

Divide each term by ( x ): [ \frac{\sin(x^2)}{x} = x  \frac{x^5}{3!} + \frac{x^9}{5!}  \frac{x^{13}}{7!} + \cdots ]

Simplify the series: [ \frac{\sin(x^2)}{x} = x  \frac{x^5}{6} + \frac{x^9}{120}  \frac{x^{13}}{5040} + \cdots ]
Therefore, the Maclaurin series for ( \frac{\sin(x^2)}{x} ) is: [ \frac{\sin(x^2)}{x} = x  \frac{x^5}{6} + \frac{x^9}{120}  \frac{x^{13}}{5040} + \cdots ]
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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