If -3-32-35-34- Find the vvalue of e i -

Θ = π/3 + π/3^2 + π/3^3 + π/3^4 +…… ∞ Above given series a GP Sum of infinite terms s = a/1 r = π/3/1 1/3 = π/3/2/3 = π/2 e^i θ = cos θ + i sin θ Substitu ...

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Standard XI

Mathematics

Euler's Representation

If θ=π/3+π/32...

Question

If $θ$ = $\frac{π}{3}$ + $\frac{π}{3^{2}}$ + $\frac{π}{3^{3}}$ + $\frac{π}{3^{4}} + \dots \dots - \infty$ , Find the vvalue of $e^{i θ}$ .

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Solution

$θ$ = $\frac{π}{3}$ + $\frac{π}{3^{2}}$ + $\frac{π}{3^{3}}$ + $\frac{π}{3^{4}} + \dots \dots - \infty$

Above given series a GP

Sum of infinite terms s = $\frac{a}{1 - r}$

= $\frac{\frac{π}{3}}{1 - \frac{1}{3}}$ = $\frac{\frac{π}{3}}{\frac{2}{3}}$ = $\frac{π}{2}$

$e^{i θ}$ = $c o s θ + i s i n θ$

Substitute $θ$ = $\frac{π}{2}$

= cos $\frac{π}{2}$ + i sin $\frac{π}{2}$

= 0+i = i

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If θ = π3 + π32 + π33 + π34+……−∞ , Find the vvalue of eiθ . ___

θ = π3 + π32 + π33 + π34+……−∞ Above given series a GP Sum of infinite terms s = a1−r = π31−13 = π323 = π2 eiθ = cosθ+isinθ Substitute θ = π2 = cosπ2 + i sin π2 = 0+i = i

If $θ$ = $\frac{π}{3}$ + $\frac{π}{3^{2}}$ + $\frac{π}{3^{3}}$ + $\frac{π}{3^{4}} + \dots \dots - \infty$ , Find the vvalue of $e^{i θ}$ .

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