Another convergence. An interval of convergence.

I have to find, for all value of constant $\displaystyle a>0$, the interval of converence of $\displaystyle $\sum_{n=0}^{\infty}\frac{x^{n}}{1+a^{n}}$$.

How do I do? By using the Ratio test I get

$\displaystyle \lim_{n \to \infty}\frac{\left (\frac{x^{n+1}}{1+a^{n+1}} \right )}{\left (\frac{x^{n}}{1+a^{n}} \right )}=\lim_{n \to \infty} \frac{x^{n+1}}{x^{n}}\frac{1+a^{n}}{1+a^{n+1}}= \lim_{n \to \infty}x\frac{1+a^{n}}{1+a^{n+1}}$

Now I will use the rule of Hopital to make this fraction 'simpler'.

$\displaystyle \lim_{n \to \infty}x\frac{1+a^{n}}{1+a^{n+1}}=x\lim_{n \to \infty}\frac{\frac{\mathrm{d} \left (1+a^{n} \right )}{\mathrm{d} n}}{\frac{\mathrm{d} \left (1+a^{n+1} \right )}{\mathrm{d} n}}=...=x\lim_{n \to \infty}a^{-1}=\frac{x}{a}$

According to the Ratio test, it states that this series is convergent if $\displaystyle \frac{x}{a}<1$, that is $\displaystyle x<a$. Divergent if $\displaystyle \frac{x}{a}>1$, that is $\displaystyle x>a$. If $\displaystyle x=a$, it has not result. If we test it by inserting the value of $\displaystyle a = 1$ on this series, it would be divergent according to Wolfram Alpha. I don't understand this part because this problem clearly states the value of a should be greater than 0, so $\displaystyle a = 1 > 0$ should also be convergent.

The other hand if we see it converges if $\displaystyle x<a$ where $\displaystyle a>0$, so I am not really sure I could say it would be $\displaystyle 0<x<a$. If it's true, the interval of convergent would be like $\displaystyle x \in ]0, a]$ as $\displaystyle a > 0$.

If I did it wrong, please tell me what to do.

Re: Another convergence. An interval of convergence.

Sorry, I won't analyze your solution and will just write what I think. According to this page, the radius of convergence is the reciprocal of $\displaystyle \lim_{n\to\infty}\left| \frac{1+a^n}{1+a^{n+1}} \right|$. If $\displaystyle a\le1$, the limit is 1, so the radius is 1. If a > 1, the limit is 1 / a, so the radius is a.

Now, what happens when |x| equals the radius? If a < 1 and x = 1, the series diverges by the integral test. If a = x = 1, the series clearly diverges. If a > 1 and x = a, again the series diverges by the integral test. It is left to check when x equals negative radius.

Re: Another convergence. An interval of convergence.

Thanks. Didn't think about it before. I have tried to check it and you were right about the values of a and r.

Generally |x| should be lesser than r, so that the series converges. If greater than r, it diverges.

The other hand |x|/a < 1 the series converges which means |x|<a or -a < x < a. So what do I have to answer more presicely?

Re: Another convergence. An interval of convergence.

You have to realise that there are infinitely many possible series that as written, your expression represents (as r and a are freely chosen). Luckily the ratio test can test all but TWO of these cases, where the ratio's limit is 1. So these two series where the ratio's limit is 1 needs to be tested with some other test as Emakarov has done :)

Re: Another convergence. An interval of convergence.

Quote:

Originally Posted by

**MathsforNewbs** The other hand |x|/a < 1 the series converges which means |x|<a or -a < x < a. So what do I have to answer more presicely?

As I said in my previous post, the radius of convergence is not always $\displaystyle a$. This is because $\displaystyle \lim_{n\to\infty} \left| x\frac{1+a^n}{1+a^{n+1}} \right|$ is not always |x| / a. And this is because one of the conditions of the l'Hôpital's rule is that the numerator and the denominator must both tend to 0 or $\displaystyle \pm\infty$. But when a < 1, $\displaystyle 1+a^n$ and $\displaystyle 1+a^{n+1}$ tend to 1, so the l'Hôpital's rule does not apply. When a < 1, the limit above is |x|, so the radius of convergence is 1. Altogether, the radius is $\displaystyle a$ when a > 1 and 1 when a ≤ 1.

Also, it is easier to see that the interval of convergence is open in all cases not by the integral test, as I said in the previous post, but because when x equals one of the interval's edges, the terms of the series don't tend to zero.