# How do you find the Maclaurin series for #arctan x# centered at x=0?

Start from the sum of a geometric series, which is:

or:

If we integrate now the series term by term we have:

At the second member we have a standard integral:

so we have:

and finally:

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To find the Maclaurin series for ( \arctan(x) ) centered at ( x = 0 ), you can use the Taylor series expansion formula. The Taylor series expansion for ( \arctan(x) ) is given by:

[ \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}x^{2n+1} ]

This series is centered at ( x = 0 ) because it's a Maclaurin series, meaning it's a Taylor series centered 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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