How to Determine the Interval of Convergence for the Given Series?

[tamtam402]

Homework Statement

I'm trying to figure out the interval of convergence of the serie:


\Sigma^{\infty}_{n=2} \frac{(-1)^{n}x^{n/2}}{nln(n)}

Homework Equations

The Attempt at a Solution

The book states that the serie isn't a power serie, but a substitution can be used to transform it into a power serie. I tried to pose y^{n} = x^{n/2}.

I also know that I must use the ratio test. Maybe there's another way to solve it, but the book states that the ratio test must be used. My problem is that I don't know how to simplify the ln(n+1) term that appears when I do that. Would anyone mind giving me a clue?

 

[DryRun]

Hi tamtam402

Please use a descriptive title related to the actual problem in your post, according to the forum rules. This will help to bring more attention to your problem.
\Sigma^{\infty}_{n=2} \frac{(-1)^{n}x^{n/2}}{nln(n)}It appears that using the Alternative Series Test might be helpful as the first step.

 

[tamtam402]

Unfortunately the book states that the problem must be done using the ratio test. I'm stuck with the ln(n+1) term, I don't know how to simplify it with the ln(n) term.

 

[DryRun]

What is x? Is it a constant? If yes, what value does it have? Is it greater than 0? You should provide all the information, or preferably the entire question. Type it all or just modify and attach a screenshot or picture to your first post.

 

[tamtam402]

x isn't a constant, it's a variable since I'm working with a power serie. I wrote all the informations given in the book in my first post.

The information given in the book is that I should do a substitution to transform the serie into a "real" power serie. Then I have to use the ratio test, and take the limit of n -> infinity on the ratio.

According to a theorem, the power serie converges for the values of x where -1 < lim n-> infinity pn < 1

The final step is to analyze the boundaries. However, I'm unable to simplify the ratio since I have a ln(n+1) term on the denominator and a ln(n) term on the numerator. I don't know how to simplify these.

 

[gopher_p]

You cannot simplify algebraically $\frac{\ln(n)}{\ln(n+1)}$. But there is a way to find $\lim_{n\rightarrow\infty}\frac{\ln(n)}{\ln(n+1)}$ using techniques from calculus. Hint: What are the limits of the numerator and denominator?

And while it's not immediately applicable, the alternating series test might prove useful later in the problem.

 

[Ray Vickson]

Homework Statement

I'm trying to figure out the interval of convergence of the serie:


\Sigma^{\infty}_{n=2} \frac{(-1)^{n}x^{n/2}}{nln(n)}

Homework Equations

The Attempt at a Solution

The book states that the serie isn't a power serie, but a substitution can be used to transform it into a power serie. I tried to pose y^{n} = x^{n/2}.

I also know that I must use the ratio test. Maybe there's another way to solve it, but the book states that the ratio test must be used. My problem is that I don't know how to simplify the ln(n+1) term that appears when I do that. Would anyone mind giving me a clue?

For small t > 0 we have ln(1+t) = t + O(t^2), so ln(n+1) = ln(n) + ln(1 + 1/n) = ln(n) + (1/n) + O(1/n^2) for large n. Using that should allow you to say enough about the ratio.

RGV