# How can I simplify this?

## Homework Statement

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

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

## 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?

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DryRun
Gold Member
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.

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
Gold Member
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.

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.

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 Helper
Dearly Missed

## Homework Statement

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

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

## 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