# What is the taylor series for #x(e^(2x))#?

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The Taylor series for ( x \cdot e^{2x} ) can be found by applying the Taylor series expansion for ( e^{2x} ) and then multiplying it term by term by the Taylor series expansion for ( x ).

The Taylor series expansion for ( e^{2x} ) is:

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

The Taylor series expansion for ( x ) is simply:

[ x = x ]

Multiplying these two series term by term gives:

[ x \cdot e^{2x} = \sum_{n=0}^{\infty} \frac{(2x)^n \cdot x}{n!} ]

This can be further simplified to:

[ x \cdot e^{2x} = \sum_{n=0}^{\infty} \frac{2^n \cdot x^{n+1}}{n!} ]

So, the Taylor series for ( x \cdot e^{2x} ) is:

[ \boxed{\sum_{n=0}^{\infty} \frac{2^n \cdot x^{n+1}}{n!}} ]

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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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