MHB What are the intervals and radius of convergence for two series with typos?

[ineedhelpnow]

find the interval of convergence of the series $\sum_{x=1}^{\infty} \frac{6x^n}{\sqrt[5]{n}}$find the radius of convergence of the series $\sum_{n=1}^{\infty} \frac{8^nx^n}{(n+5)^2}$

 

[ineedhelpnow]

for the first one i think you use the integral test and for the second the ratio test but i don't understand HOW you use them to determine the interval and radius?

 

[evinda]

find the interval of convergence of the series $\sum_{x=1}^{\infty} \frac{6x^n}{\sqrt[5]{n}}$

$$\xi=0$$

$$\rho=\lim_{n \to +\infty} \sqrt[n]{|a_n|}=\lim_{n \to +\infty} \sqrt[n]{\frac{6}{\sqrt[5]{n}}}=\lim_{n \to +\infty} \frac{6^{\frac{1}{n}}}{n^{\frac{1}{5n}}}=1$$

$$R=\frac{1}{ \rho}=1$$

So,the series converges absolutely for $x \in (-1,1)$ and diverges for $x \notin [-1,1]$

Now you have to check the convergence at the points $-1$and $1$.

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find the radius of convergence of the series $\sum_{n=1}^{\infty} \frac{8^nx^n}{(n+5)^2}$

$$\xi=0$$

$$\rho=\lim_{n \to +\infty} \sqrt[n]{|a_n|}=\lim_{n \to +\infty} \sqrt[n]{\frac{8^n}{(n+5)^2}}=\lim_{n \to +\infty} \frac{8}{(n+5)^{\frac{2}{n}}}=8$$So,the radius of convergence is $R=\frac{1}{ \rho}=\frac{1}{8}$ .

 

[ineedhelpnow]

thanks! what about the x though?

 

[evinda]

thanks! what about the x though?

Oh,sorry! (Blush) I thought that it would be $\sum_{n=1}^{\infty} \frac{6x^n}{\sqrt[5]{n}}$.
When it is like that: $\sum_{x=1}^{\infty} \frac{6x^n}{\sqrt[5]{n}}$ , the ratio test cannot be used!

 

[ineedhelpnow]

(Tmi) do you know how to do it with x?

 

[I like Serena]

(Tmi) do you know how to do it with x?

Hi!

What do you get if you apply the ratio test?

In other words, what is:
$$L=\lim_{n\to \infty} \left| \frac{\dfrac{6x^{n+1}}{\sqrt[5]{n+1}}}{\dfrac{6x^n}{\sqrt[5]{n}}} \right|$$

 

[evinda]

find the interval of convergence of the series $\sum_{x=1}^{\infty} \frac{6x^n}{\sqrt[5]{n}}$

Is it maybe a typo and the series is $\sum_{n=1}^{\infty} \frac{6x^n}{\sqrt[5]{n}}$ ? (Thinking)

 

[ineedhelpnow]

that actually is a typo but i figured them out. thanks though