Showing that $|z^{n+1}-z^n|=|z-1|$












-2












$begingroup$


I need to show that $|z^{n+1}-z^n|=|z-1|$ where $z$ is a complex number, and the series ${ a_{n} } ^infty _{n=1}$ is defined as $a_n=z^n$. It is assumed that $|z|=1$.



I'm having trouble showing that :/






sequences-and-series complex-numbers







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  • 2




    $begingroup$
    $$|ab|=|a||b|$$
    $endgroup$
    – lab bhattacharjee
    Jan 5 at 10:06
















-2












$begingroup$


I need to show that $|z^{n+1}-z^n|=|z-1|$ where $z$ is a complex number, and the series ${ a_{n} } ^infty _{n=1}$ is defined as $a_n=z^n$. It is assumed that $|z|=1$.



I'm having trouble showing that :/






sequences-and-series complex-numbers







share|cite|improve this question









$endgroup$








  • 2




    $begingroup$
    $$|ab|=|a||b|$$
    $endgroup$
    – lab bhattacharjee
    Jan 5 at 10:06














-2












-2








-2





$begingroup$


I need to show that $|z^{n+1}-z^n|=|z-1|$ where $z$ is a complex number, and the series ${ a_{n} } ^infty _{n=1}$ is defined as $a_n=z^n$. It is assumed that $|z|=1$.



I'm having trouble showing that :/






sequences-and-series complex-numbers







share|cite|improve this question









$endgroup$




I need to show that $|z^{n+1}-z^n|=|z-1|$ where $z$ is a complex number, and the series ${ a_{n} } ^infty _{n=1}$ is defined as $a_n=z^n$. It is assumed that $|z|=1$.



I'm having trouble showing that :/






sequences-and-series complex-numbers




sequences-and-series complex-numbers






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 5 at 10:04









Kasper LarsenKasper Larsen

113




113








  • 2




    $begingroup$
    $$|ab|=|a||b|$$
    $endgroup$
    – lab bhattacharjee
    Jan 5 at 10:06














  • 2




    $begingroup$
    $$|ab|=|a||b|$$
    $endgroup$
    – lab bhattacharjee
    Jan 5 at 10:06








2




2




$begingroup$
$$|ab|=|a||b|$$
$endgroup$
– lab bhattacharjee
Jan 5 at 10:06




$begingroup$
$$|ab|=|a||b|$$
$endgroup$
– lab bhattacharjee
Jan 5 at 10:06










1 Answer
1






active

oldest

votes


















2












$begingroup$

Note that
$$|z^{n+1} - z^n| = |z^n(z - 1)| = |z^n| cdot |z - 1| = |z|^n cdot |z - 1| = |z - 1|.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I can't believe I didn't see that. Thank you.
    $endgroup$
    – Kasper Larsen
    Jan 5 at 10:10











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1 Answer
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active

oldest

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









2












$begingroup$

Note that
$$|z^{n+1} - z^n| = |z^n(z - 1)| = |z^n| cdot |z - 1| = |z|^n cdot |z - 1| = |z - 1|.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I can't believe I didn't see that. Thank you.
    $endgroup$
    – Kasper Larsen
    Jan 5 at 10:10
















2












$begingroup$

Note that
$$|z^{n+1} - z^n| = |z^n(z - 1)| = |z^n| cdot |z - 1| = |z|^n cdot |z - 1| = |z - 1|.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I can't believe I didn't see that. Thank you.
    $endgroup$
    – Kasper Larsen
    Jan 5 at 10:10














2












2








2





$begingroup$

Note that
$$|z^{n+1} - z^n| = |z^n(z - 1)| = |z^n| cdot |z - 1| = |z|^n cdot |z - 1| = |z - 1|.$$






share|cite|improve this answer









$endgroup$



Note that
$$|z^{n+1} - z^n| = |z^n(z - 1)| = |z^n| cdot |z - 1| = |z|^n cdot |z - 1| = |z - 1|.$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 5 at 10:07









Theo BenditTheo Bendit

17.2k12149




17.2k12149












  • $begingroup$
    I can't believe I didn't see that. Thank you.
    $endgroup$
    – Kasper Larsen
    Jan 5 at 10:10


















  • $begingroup$
    I can't believe I didn't see that. Thank you.
    $endgroup$
    – Kasper Larsen
    Jan 5 at 10:10
















$begingroup$
I can't believe I didn't see that. Thank you.
$endgroup$
– Kasper Larsen
Jan 5 at 10:10




$begingroup$
I can't believe I didn't see that. Thank you.
$endgroup$
– Kasper Larsen
Jan 5 at 10:10


















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