Problem in finding the radius of convergence of a series

In summary, the conversation discusses the concept of 2^(n/2) and 3^(n/3) and why 2^(n/2) is considered little o of 3^(n/3). The solution provided involves finding the radius and using the formula [1+k^n+n^7/3^(n/3)]^(1/n) to show that as n approaches infinity, the result converges to 1, proving that 2^(n/2) is indeed less than 3^(n/3).
  • #1
Amaelle
310
54
Homework Statement
look at the image
Relevant Equations
asymptotic behaviour, raduis of convergence
Good day
1612550922783.png

I'm trying to find the radius of this serie, and here is the solution
1612551034715.png

I just have problem understanding why 2^(n/2) is little o of 3^(n/3) ?

many thanks in advance

Best regards!
 
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  • #2
Amaelle said:
why 2^(n/2) is little o of 3^(n/3)
$$2^{n/2}= 1.4142^n \qquad 3^{n/3}= 1.4442^n \quad \Rightarrow \quad 2^{n/2} < 3^{n/3}$$
 
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  • #3
[tex][2^{n/2}+3^{n/3}+n^7]^{1/n}=3^{1/3}[1+k^n+\frac{n^7}{3^{n/3}}]^{1/n}[/tex]
where
[tex]k=\frac{2^{1/2}}{3^{1/3}}[/tex]
[tex]k^6=\frac{8}{9}<1[/tex]
so k<1. As n##\rightarrow +\infty##
[tex][1+k^n+\frac{n^7}{3^{n/3}}]^{1/n} \rightarrow 1[/tex]
 
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  • #4
anuttarasammyak said:
[tex][2^{n/2}+3^{n/3}+n^7]^{1/n}=3^{1/3}[1+k^n+\frac{n^7}{3^{n/3}}]^{1/n}[/tex]
where
[tex]k=\frac{2^{1/2}}{3^{1/3}}[/tex]
[tex]k^6=\frac{8}{9}<1[/tex]
so k<1. As n##\rightarrow +\infty##
[tex][1+k^n+\frac{n^7}{3^{n/3}}]^{1/n} \rightarrow 1[/tex]
So beautifully explained!, thanks a million!
 
  • #5
BvU said:
$$2^{n/2}= 1.4142^n \qquad 3^{n/3}= 1.4442^n \quad \Rightarrow \quad 2^{n/2} < 3^{n/3}$$
Nice shot! thanks a million!
 
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Related to Problem in finding the radius of convergence of a series

1. What is the radius of convergence of a series?

The radius of convergence of a series is a value that represents the distance from the center of the series where the series will converge. It is usually denoted by the letter R.

2. Why is it important to find the radius of convergence of a series?

Knowing the radius of convergence of a series is important because it tells us the values of x for which the series will converge. This allows us to determine the convergence or divergence of the series and to accurately calculate the value of the series at specific points.

3. How do you find the radius of convergence of a series?

The most common method for finding the radius of convergence is by using the ratio test. This involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If the limit is less than 1, the series will converge. The value of x at which the limit is equal to 1 is the radius of convergence.

4. Can the radius of convergence of a series be negative?

No, the radius of convergence cannot be negative. It represents a distance and therefore must be a positive value. If the radius of convergence is negative, it is likely that an error has been made in the calculation or the series does not converge.

5. What happens if the radius of convergence of a series is infinite?

If the radius of convergence is infinite, it means that the series will converge for all values of x. This is also known as a power series. In this case, the series can be used to represent a function and can be manipulated to find the values of the function at different points.

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