Let- fp-ei-p2- e2i-p2-e3i-p2-e4i-p2- ei-p- where i-1 and p - N then limn - fn- is

Given: f_p(α)=e^iαp^2. e^2iαp^2.e^3iαp^2.e^4iαp^2⋯ e^iαp ⇒ f_p(α )=e^iαp^2(1+2+3+4+… +p) ⇒ f_p(α )=e^iαp^2.p(p+1)2 ⇒ f_p(α )=e^iα2 ( 1+1p ) ∴lim_n → nbsp;∞ f ...

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Byju's Answer

Standard XII

Mathematics

Euler's Representation

Let, fpα=eiαp...

Question

Let $f_{p} (α) = e^{\frac{i α}{p^{2}}} . e^{\frac{2 i α}{p^{2}}} . e^{\frac{3 i α}{p^{2}}} . e^{\frac{4 i α}{p^{2}}} \dots e^{\frac{i α}{p}}$ , (where $i = \sqrt{- 1}$ and $p \in N$ ) then ${lim}_{n \to \infty} f_{n} (π)$ is

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Solution

The correct option is C
i

$f_{p} (α) = e^{\frac{i α}{p^{2}} (1 + 2 + 3 + 4 + \dots + p)} = e^{\frac{i α}{p^{2}} . \frac{p (p + 1)}{2}} = e^{\frac{i α}{2} (1 + \frac{1}{p})} ∴ {lim}_{n \to \infty} f_{n} (π) = {lim}_{n \to \infty} e^{\frac{i π}{2} (1 + \frac{1}{n})} = e^{\frac{i π}{2} (1 + 0)} = e^{\frac{i π}{2}} = cos \frac{π}{2} + i sin \frac{π}{2} = i$

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Similar questions

Let

f_{p} (β) = (cos \frac{β}{p^{2}} + i sin \frac{β}{p^{2}}) + (cos \frac{2 β}{p^{2}} + i sin \frac{2 β}{p^{2}}) \dots \dots (cos \frac{β (p - 1)}{p^{2}} + i sin \frac{β (p - 1)}{p^{2}}) + (cos \frac{β}{p} + i sin \frac{β}{p})

then

lim n \to \infty 1 / f_{n} (π) =

Q. Let

f_{p} (a) = e^{\frac{i a}{p^{2}}} \cdot e^{\frac{2 i a}{p^{2}}} \cdot e^{\frac{3 i a}{p^{2}}} \dots \cdot e^{\frac{i a}{p}}

(Where

i = \sqrt{- 1} and p \in N) then {lim}_{n \to \infty} f_{n} (π)

Q. Let

f_{p} (β) = (cos \frac{β}{p^{2}} + i sin \frac{β}{p^{2}}) (cos \frac{2 β}{p^{2}} + i sin \frac{2 β}{p^{2}}) . . . . (cos \frac{β}{p} + i sin \frac{β}{p})

then

lim n \to \infty f_{n} (π) = . . . . .

Q. Let

P

be a non-singular matrix and

I + P + P^{2} + \dots + P^{n} = O

. Then

P^{- 1}

is equal to (where

n \in Z^{+}

)

Q. Let p be a nonsingular matrix, and

I + p + p^{2} + . . . . . + p^{n} = 0

, then find

p^{- 1}

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Let fp(α)=eiαp2.e2iαp2.e3iαp2.e4iαp2…eiαp, (where i=√−1 and p∈N) then limn→∞fn(π) is

The correct option is C i fp(α)=eiαp2(1+2+3+4+…+p)=eiαp2.p(p+1)2=eiα2(1+1p)∴ limn→∞fn(π)=limn→∞eiπ2(1+1n)=eiπ2(1+0)=eiπ2=cosπ2+i sinπ2=i

Let $f_{p} (α) = e^{\frac{i α}{p^{2}}} . e^{\frac{2 i α}{p^{2}}} . e^{\frac{3 i α}{p^{2}}} . e^{\frac{4 i α}{p^{2}}} \dots e^{\frac{i α}{p}}$ , (where $i = \sqrt{- 1}$ and $p \in N$ ) then ${lim}_{n \to \infty} f_{n} (π)$ is