# How do you find the maclaurin series expansion of #f(x) = x^2 arctan (x^3)#?

Alternatively, compactly

#sum_{i=0}^infty (-1)^{n} (x^3)^{2n+1}/{2n+1}= sum_{i=0}^infty (-1)^{n} x^{6n+3}/{2n+1}#

#sum_{i=0}^infty (-1)^{n} x^2*x^{6n+3}/{2n+1}= sum_{i=0}^infty (-1)^{n} x^{6n+5}/{2n+1}#

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To find the Maclaurin series expansion of ( f(x) = x^2 \arctan(x^3) ), you would follow these steps:

- Express the function as a series using known Maclaurin series expansions.
- Find the derivatives of the function.
- Evaluate the derivatives at ( x = 0 ).
- Write the Maclaurin series expansion using the coefficients obtained.

The Maclaurin series expansion of ( f(x) = x^2 \arctan(x^3) ) is:

[ f(x) = x^2 \arctan(x^3) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n ]

where ( f^{(n)}(0) ) denotes the ( n )-th derivative of ( f(x) ) evaluated at ( x = 0 ).

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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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