# How do you do the taylor series expansion of #arctan(x)# and #xsinx#?

Knowing that performing operations on a Taylor series parallels performing operations on the function which the series represents, we can start from here and transform the series through a sequence of operations.

We should consider the properties of limits with sums:

Meaning that we can integrate sums term by term:

That's all!

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The Taylor series expansion of arctan(x) is given by:

arctan(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + ...

The Taylor series expansion of x*sin(x) is given by:

x*sin(x) = x^2/1! - x^4/3! + x^6/5! - x^8/7! + ...

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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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