EaseUS Data Recovery Wizard V16.9 Keygen Serial Key Keygen [UPD]

EaseUS Data Recovery Wizard V16.9 Keygen Serial Key Keygen [UPD]

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EaseUS Data Recovery Wizard V16.9 Keygen Serial Key Keygen

Easeus Data Recovery Wizard v16.9 Keygen Serial Key keygenQ:

Prove$z_n$ converges to $z$ and $|z_n| \leq |z|$ for all $n$

I was doing this problem

Let $z\in\mathbb{C}$. Define $z_n = z^n + z^{2n} +… +z^{n^2}$.
Prove that $z_n$ converges to $z$ and that $|z_n| \leq |z|$ for all $n$.

Hint: Let $|z| = d$, and let $d_n = |z^n|$. Then $d = d_1$, and $d_n \leq d$, so $\limsup d_n \leq d$.
Then it says use the Comparison Test to show that $d_n$ approaches $d$.
I don’t really understand what they mean by ” $d_n \leq d$, so $\limsup d_n \leq d$ “. I tried to show $d_n \leq d$ but it seems that $d_n \geq d$, and I couldn’t show $d_n \leq d$ by just using the hint.
I know $|z_n| \leq |z|$ because $|z_n| = |z^n + z^{2n} +… +z^{n^2}| \leq |z|^n \leq |z|$
and $z_n$ converges to $z$ because $|z^n| \to 0$ so $z^n = o(1)$. But still I can’t use the hint.
Can anyone explain the hint to me?
Thank you.

A:

I assume you are told that $\lvert z^n\rvert\le\lvert z\rvert$ for every $n$.
If $d_n$ is the distance of $z_n$ to $0$, then
$$d_n\le\lvert z\rvert\lvert z^n\rvert\le d_n\tag{1}$$
and therefore
$$d_n\

https://colab.research.google.com/drive/1UruKnphDgYpZIMLjlLel1ftfYS-xfNgr
https://ello.co/enuntruca/post/ahkekd4piislbh3fp4i8gq
https://ello.co/maylitherea/post/pfkd2iq3qsgxbjd0oomgzw
https://documenter.getpostman.com/view/21828392/UzXUPu2A
https://ello.co/9comlaqcomne/post/gndqge66obf-4ou3-x4eoa
https://documenter.getpostman.com/view/21882342/UzdtVnh6
https://ello.co/7allagista/post/tt93le8x2wt190qphlw0aw
https://ello.co/8orprobpcris-to/post/sy6g37jp2-9x53wi434wfg
https://colab.research.google.com/drive/1SlFYJT1NR6LknxG3MSF-sJzbEqN4pR91
https://ello.co/diascurxsaun/post/nioxui6tg10blyaeswtbgq

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