How do you find the radius of convergence of #sum_(n=0)^oox^n# ?
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To find the radius of convergence of the series ( \sum_{n=0}^\infty x^n ), where ( x ) is a real or complex number, you can use the ratio test:

Apply the ratio test: [ \lim_{n \to \infty} \left \frac{a_{n+1}}{a_n} \right = \lim_{n \to \infty} \left \frac{x^{n+1}}{x^n} \right = \lim_{n \to \infty} x ]

Determine the convergence behavior based on the value of ( x ):
 If ( x < 1 ), the series converges absolutely.
 If ( x > 1 ), the series diverges.
 If ( x = 1 ), the test is inconclusive, and we need to investigate further.

Thus, the radius of convergence, denoted by ( R ), is the distance from the center of the series (which is 0 in this case) to the nearest point at which the series converges absolutely. For the series ( \sum_{n=0}^\infty x^n ), ( R = 1 ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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