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

#sum_(n=0)^oo (-1)^n x^n# with radius of convergence#1#

We choose each successive term to cancel out the extraneous term left over by the previous ones.

Then writing it out formally...

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To find a power series representation for ( f(x) = \frac{1}{1+x} ), you can use the geometric series formula:

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

This is valid for ( |x| < 1 ). So, the power series representation for ( f(x) ) is:

[ f(x) = \sum_{n=0}^{\infty} (-1)^n x^n ]

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

[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(-1)^{n+1} x^{n+1}}{(-1)^n x^n} \right| = \lim_{n \to \infty} |x| ]

Since the limit must be less than 1 for the series to converge, the radius of convergence is ( R = 1 ). Therefore, the power series representation for ( f(x) ) is valid for ( |x| < 1 ).

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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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