Verifying an asymptotic sequence

Question

I want to verify that

i) $\phi_n (z) = ln(1+z^n)$ as $z \rightarrow 0$

ii) $\phi_n (z) = z^ne^{-nz}$ as $z \rightarrow \infty$

are both asymptotic sequences.

The term 'verify' somewhat confuses me in terms of what would be considered sufficient validation.

For question i) I was asked previously to find the order for $ln(1+z)$ as $z \rightarrow 0$ and found: $ln(1+z)= z + \mathcal{O}(z^2)$

Does this mean automatically it is an asymptotic sequence and thus imply the same to be true for i) ?

In regards to ii) I'm not entirely sure how to go about 'veryifing' it is an asymptotic sequence. Is there a sort of routine check I could do?

All help is appreciated.

Answer 1

For your first sequence, $\log(1+z^n) = z^n + O(z^{2n})$ as $z \to 0$, so $\log(1+z^{n+1}) = o(\log(1+z^n))$.

For your second, $\dfrac{\phi_{n+1}(z)}{\phi_n(z)} = z e^{-z} \to 0$ as $z \to +\infty$.

Answer 2

You should prove that there exist two constants $\Gamma$ and $\beta$ that as $z \rightarrow \infty, \ z^ne^{-nz}\lt \Gamma e^{-\beta z}$