# How do you find a power series representation for #f(x)=xln(x+1)# and what is the radius of convergence?

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To find a power series representation for (f(x) = x \ln(x + 1)), we can start by recognizing that (\ln(x + 1)) has a known power series representation:

[\ln(1 + z) = z - \frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} + \cdots]

Substitute (z = x) into this series, then multiply the resulting series by (x). This gives us the power series representation for (x \ln(x + 1)).

The radius of convergence of this series can be found using the ratio test:

[R = \lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}} \right|]

Where (a_{n}) is the coefficient of the nth term in the power series representation.

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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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