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Question Number 183536 by SANOGO last updated on 26/Dec/22

radius of convergence of serie:  Σ_(nεN) ((cos(((3π)/3)))/5^n )z^n

$${radius}\:{of}\:{convergence}\:{of}\:{serie}: \\ $$$$\underset{{n}\epsilon{N}} {\sum}\frac{{cos}\left(\frac{\mathrm{3}\pi}{\mathrm{3}}\right)}{\mathrm{5}^{{n}} }{z}^{{n}} \\ $$

Commented by mr W last updated on 26/Dec/22

strange!  why do you write ((3π)/3)? not directly π?  cos (((3π)/3))=cos π=−1

$${strange}! \\ $$$${why}\:{do}\:{you}\:{write}\:\frac{\mathrm{3}\pi}{\mathrm{3}}?\:{not}\:{directly}\:\pi? \\ $$$$\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{3}}\right)=\mathrm{cos}\:\pi=−\mathrm{1} \\ $$

Answered by aleks041103 last updated on 27/Dec/22

=−Σ_(n=1) ^∞ ((z/5))^n   this is a geometric series.  to converge, ∣(z/5)∣<1  ⇒∣z∣<5

$$=−\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{{z}}{\mathrm{5}}\right)^{{n}} \\ $$$${this}\:{is}\:{a}\:{geometric}\:{series}. \\ $$$${to}\:{converge},\:\mid\frac{{z}}{\mathrm{5}}\mid<\mathrm{1} \\ $$$$\Rightarrow\mid{z}\mid<\mathrm{5} \\ $$

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