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

Answer 1

#(-5, 13]#

Compare with # ln(1+X)=sum(-1)^(n-1)X^n/n, n=1,, 2, 3, ..#,
# -1 < X <=1#.
Easily, #X =(x-4)/9#.
So,# -1< (x-4)/9 <=1#.
And so, explicitly, #-5 < x <=13#.
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Answer 2

#sum_{n=1}^{oo} ((x − 4)^n)/(n(−9)^n) =-log_e abs((5 + x)/9)#

is convergent for #-5 < x < 13#

#sum_{n=1}^{oo} ((x − 4)^n)/(n(−9)^n) =sum_{n=0}^{oo}(((x-4)/(-9))^n)/n #
For #abs y < 1#
#d/(dy)sum_{k=1}^{oo}(y^k)/k = sum_{k=1}^{oo}y^{k-1}=1/(1-y)#

so

#sum_{k=1}^{oo}(y^k)/k=-log_e abs(1-y)#
then for #abs((x-4)/(-9))<1 -> -5 < x <13# the series is convergent
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Answer 3

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:

  1. If ( L < 1 ), the series converges absolutely.
  2. If ( L > 1 ), the series diverges.
  3. 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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Answer from HIX Tutor

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.

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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