What is the interval of convergence of #sum (n^3)(x^(3n))/(3^(3n)) #?

Answer 1

#(-3, 3)#

If #x in (-3, 3)#, then #abs(x^3/3^3) < 1#, so #1 - abs(x^3/3^3) > 0#
Let #delta = 1 - abs(x^3/3^3) > 0#
Let #N = ceil(7/delta)#
Suppose #n >= 1#:
#(n+1)^3/n^3 = (n^3+3n^2+3n+1)/n^3#
#= 1+3/n+3/n^2+1/n^3 <= 1+7/n#
So if #n >= N#, then #(n+1)^3/n^3 <= 1+7/ceil(7/delta) <= 1+delta#
So #(n+1)^3/n^3 abs(x^3/3^3) <= (1+delta)(1-delta) = 1-delta^2 < 1#
Then from #N# onwards, the series converges faster than a geometric series with common ratio #1-delta^2#.
On the other hand, if #abs(x) >= 3#, then the series diverges faster than a geometric series with common ratio #1#.
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Answer 2

To find the interval of convergence of the series (\sum \frac{n^3 x^{3n}}{3^{3n}}), 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{(n+1)^3 x^{3(n+1)}}{3^{3(n+1)}} \cdot \frac{3^{3n}}{n^3 x^{3n}} \right| ]

Simplify the expression:

[ \lim_{{n \to \infty}} \left| \frac{(n+1)^3}{3^3 n^3} \cdot \frac{x^3}{x^3} \right| ]

Take the limit:

[ \lim_{{n \to \infty}} \left| \frac{(n+1)^3}{3^3 n^3} \right| |x|^3 = \frac{|x|^3}{3^3} ]

For the series to converge, this limit must be less than 1:

[ \frac{|x|^3}{3^3} < 1 ]

Solve for (|x|):

[ |x|^3 < 3^3 \implies |x| < 3 ]

Thus, the interval of convergence is ((-3, 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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