# What is the interval of convergence of #sum ((x − 4)^n)/(n*(−9)^n)#?

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is convergent for

so

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To find the interval of convergence of the series ( \sum \frac{(x - 4)^n}{n \cdot (-9)^n} ), we can use the ratio test.

The ratio test states that for a series ( \sum a_n ), if ( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L ), then:

- If ( L < 1 ), the series converges absolutely.
- If ( L > 1 ), the series diverges.
- If ( L = 1 ), the test is inconclusive.

Let's apply the ratio test to our series:

[ a_n = \frac{(x - 4)^n}{n \cdot (-9)^n} ] [ a_{n+1} = \frac{(x - 4)^{n+1}}{(n+1) \cdot (-9)^{n+1}} ]

[ \frac{a_{n+1}}{a_n} = \frac{\frac{(x - 4)^{n+1}}{(n+1) \cdot (-9)^{n+1}}}{\frac{(x - 4)^n}{n \cdot (-9)^n}} = \frac{(x - 4)(-9)}{n+1} ]

[ \lim_{n \to \infty} \left| \frac{(x - 4)(-9)}{n+1} \right| = \lim_{n \to \infty} \frac{|x - 4| \cdot 9}{n+1} = 0 ]

Since the limit is 0 for all ( x ), the ratio test tells us that the series converges for all real numbers ( x ). Therefore, the interval of convergence is ( (-\infty, +\infty) ).

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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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