# How do you find the Taylor polynomial of degree 10 of the function #arctan(x^3)# at a = 0?

Start from the sum of the geometric series:

for

Let

converging for

In the interval of convergence we can integrate term by term:

Let now

and truncating at the third term:

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To find the Taylor polynomial of degree 10 of the function arctan(x^3) at (a = 0), we can use the Taylor series expansion formula for arctan(x). The Taylor series expansion for arctan(x) is given by:

[ \arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \frac{x^{11}}{11} + \ldots ]

Substituting (x^3) for (x) in the series, we get:

[ \arctan(x^3) = x^3 - \frac{(x^3)^3}{3} + \frac{(x^3)^5}{5} - \frac{(x^3)^7}{7} + \frac{(x^3)^9}{9} - \frac{(x^3)^{11}}{11} + \ldots ]

[ = x^3 - \frac{x^9}{3} + \frac{x^{15}}{5} - \frac{x^{21}}{7} + \frac{x^{27}}{9} - \frac{x^{33}}{11} + \ldots ]

The Taylor polynomial of degree 10 of the function arctan(x^3) at (a = 0) is obtained by considering the terms up to (x^{10}) in the series expansion. Thus, the Taylor polynomial of degree 10 is:

[ P_{10}(x) = x^3 - \frac{x^9}{3} + \frac{x^{15}}{5} - \frac{x^{21}}{7} + \frac{x^{27}}{9} ]

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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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