State whether the following statement is true or false.If Zr=cos3rπ+isin3rπ,r=1,2,3,...... then z1z2z3........∞=i

z1=cos3π+isin3π=ei3πz2=cos32π+isin32π=ei32πz3=cos33π+isin33π=ei33πSo,z1z2z3.....∞=eiπ/3eiπ/32eiπ/33....∞=eiπ[1/3+1/32+1/33+.....∞]=eiπ/2=cos(π/2)+isin(π/2)=0+i=i

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State whether the following statement is true or false. If Z_r

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State whether the following statement is true or false.
If $Z_{r} = cos \frac{π}{3 ^{r}} + i s in \frac{π}{3 ^{r}}, r = 1, 2, 3, ......$ then $z_{1} z_{2} z_{3} ........\infty = i$

True

False

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State whether the following statement is true or false.

Z_{r} = cos \frac{π}{3 ^{r}} + i s in \frac{π}{3 ^{r}}, r = 1, 2, 3, ......

then

z_{1} z_{2} z_{3} ........\infty = i

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Text solutionVerified

z_{1} = cos \frac{π}{3} + i s in \frac{π}{3} = e^{i \frac{π}{3}}

z_{2} = cos \frac{π}{3 ^{2}} + i s in \frac{π}{3 ^{2}} = e^{i \frac{π}{3 ^{2}}}

z_{3} = cos \frac{π}{3 ^{3}} + i s in \frac{π}{3 ^{3}} = e^{i \frac{π}{3 ^{3}}}

So,

z_{1} z_{2} z_{3} .....\infty = e^{iπ /3} e^{iπ / 3^{2}} e^{iπ / 3^{3}} ....\infty = e^{iπ [1/3 + 1/ 3^{2} + 1/ 3^{3} + .....\infty]} = e^{iπ /2} = cos (π /2) + i s in (π /2) = 0 + i = i

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Question Text	State whether the following statement is true or false. If $Z_{r} = cos \frac{π}{3 ^{r}} + i s in \frac{π}{3 ^{r}}, r = 1, 2, 3, ......$ then $z_{1} z_{2} z_{3} ........\infty = i$
Answer Type	Text solution:1
Upvotes	150

State whether the following statement is true or false.If Zr​=cos3rπ​+isin3rπ​,r=1,2,3,...... then z1​z2​z3​........∞=i

Text solutionVerified

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State whether the following statement is true or false.
If $Z_{r} = cos \frac{π}{3 ^{r}} + i s in \frac{π}{3 ^{r}}, r = 1, 2, 3, ......$ then $z_{1} z_{2} z_{3} ........\infty = i$