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Sommes infinies<br />

Définition :<br />

∞<br />

i=0<br />

zi = lim<br />

n→∞<br />

n<br />

zi.<br />

i=0<br />

Théorème : Si |z| < 1, alors<br />

Démonstration :<br />

∞<br />

i=0<br />

∞<br />

i=0<br />

z i = 1<br />

1 − z .<br />

z i = lim<br />

n→∞<br />

= lim<br />

n→∞<br />

=<br />

n<br />

z i<br />

i=0<br />

1<br />

1 − z .<br />

1 − z n+1<br />

1 − z<br />

310

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