I am preparing for my exam and need help with the following tasks:
Let a sequence be defined as $\displaystyle\sum_{n=1}^{\infty} \frac{x}{1+n^{\alpha}x^{\beta}}$ with $\alpha$ and $\beta$ be real constants.
- Show that the sequence converges for each $x\in (0,1]$ iff $\alpha$>1.
- Let $\alpha>1$ and $\beta \leq 1$. Prove that the convergence in 1. is uniform.
- Let $\alpha>\beta >1$. Prove that the convergence in 1. is uniform. (Hint: Calculate the supremum of the summand in [0,1]
Well this is what I have (even though its not much).
- For task 1, we probably have to show that the radius of convergence is 1 and thus the series converges for |x|<1. Unfortunately I don't know how to prove that. Applying the ratio test, I get $\lim\limits_{n\to\infty} $ $\frac{1+n^{\alpha}x^{\beta}}{1+(n+1)^{\alpha}x^{\beta}}$, but what do I have to do now? I mean isn't the limit always 0 for all $\alpha>0$, since the denominator is bigger than the numerator? If we get |x|<1, I would then proof if the series is convergent for x=1. Since $\displaystyle\sum_{n=1}^{\infty} \frac{1}{1+n^{\alpha}}\leq \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{\alpha}}$. Thus the series is converging for $\alpha$>1.
- For task 2 I am a bit confused that we have to prove that the "convergence in 1" is uniformly convergent.... Well what I would do is to prove this for $\beta =1$ at first. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{1+n^{\alpha}}$ . We know that it is converging.
Lets say $\beta$$<$$1$. Thus we can write $\displaystyle\sum_{n=1}^{\infty} \frac{x^{1-\beta}}{1+n^{\alpha}}$ with $1-\beta$$>0$. If $x=1$ we know it is converging. The same goes for $0<x<1$, we proved that in 1. But how do we calculate the limit of the series so that we can check uniform convergence
- Same problem goes for task 3. I just don't have any idea how to solve this.
Is there anyone who could give me an advice? I would be very grateful.