How do you find the interval of convergence #Sigma (3^n(x-4)^(2n))/n^2# from #n=[1,oo)#?

Answer 1

#4-1/sqrt3 <= x <= 4+1/sqrt3#

Where #a_n=(3^n(x-4)^(2n))/n^2#, find the ratio #a_(n+1)/a_n#:
#a_(n+1)/a_n=(3^(n+1)(x-4)^(2n+2))/(n+1)^2*n^2/(3^n(x-4)^(2n))#
#=3^(n+1)/3^n((x-4)^(2n+2)/(x-4)^(2n))n^2/(n+1)^2#
#=3(x-4)^2(n/(n+1))^2#

And:

#lim_(nrarroo)abs(a_(n+1)/a_n)=lim_(nrarroo)abs(3(x-4)^2(n/(n+1))^2)#
#=3(x-4)^2lim_(nrarroo)abs((n/(n+1))^2)#
The limit approaches #1#:
#=3(x-4)^2#
Through the ratio test, the series converges when #lim_(nrarroo)abs(a_(n+1)/a_n)<1#:
#3(x-4)^2<1#
#-1/sqrt3 < x-4 < 1/sqrt3#
#4-1/sqrt3 < x < 4+1/sqrt3#

Test the intervals to see if the integral converges at the extremes:

At #x=4-1/sqrt3#, the series is:
#sum_(n=1)^oo(3^n((4-1/sqrt3)-4)^(2n))/n^2=sum_(n=1)^oo(3^n(-1/sqrt3)^(2n))/n^2#
#=sum_(n=1)^oo(3^n(-1)^(2n)(1/sqrt3)^(2n))/n^2=sum_(n=1)^oo(3^n(1/3^n))/n^2=sum_(n=1)^oo1/n^2#
Which converges through the p-series test, so #x=4-1/sqrt3# is included in the interval of convergence.
At #x=4+1/sqrt3#:
#sum_(n=1)^oo(3^n((4+1/sqrt3)-4)^(2n))/n^2=sum_(n=1)^oo(3^n(1/sqrt3)^(2n))/n^2=sum_(n=1)^oo1/n^2#

Which converges as well. So the interval of convergence is:

#4-1/sqrt3 <= x <= 4+1/sqrt3#
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Answer 2

To find the interval of convergence for the series ∑(3^n(x-4)^(2n))/n^2 from n=1 to infinity, we will apply the ratio test.

First, we express the general term of the series:

a_n = (3^n(x-4)^(2n))/n^2

Next, we apply the ratio test:

lim (n→∞) |a_{n+1}/a_n|

= lim (n→∞) |[3^(n+1)(x-4)^(2(n+1))/((n+1)^2)] * [(n^2)/(3^n(x-4)^(2n))]|

= lim (n→∞) |(3(x-4)^2)/(n+1)^2|

To determine convergence, we take the limit:

lim (n→∞) |(3(x-4)^2)/(n+1)^2|

Since the limit depends on n, we evaluate it as n approaches infinity:

= |3(x-4)^2/∞|

= 0

Since the limit is less than 1, the series converges for all real numbers x.

Thus, the interval of convergence is (-∞, ∞).

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Answer 3

The interval of convergence for the given series is (|x - 4| < 3).

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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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