Let $$fp($$α$$)=ei$$α$$p2$$⋅$$e2i$$α$$p2$$…….eiαp, p∈N where $$i=$$−$$1$$ , then find the value of limn→∞ $$fn($$π$$)$$

$$fp($$α$$)=ei$$α$$p2(1+2+$$…$$+p)=ei$$α$$21+1p$$∴limn→∞ $$fn($$π$$)=limn$$→∞ ei$$π$$$$21+1n=ei$$$$π$$$$2$$⇒limn→∞ $$fn($$π$$)=ei$$$$π$$$$2=1$$

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$Let f_{p} (α) = e^{\frac{i α}{p^{2}}} \cdot e^{\frac{2 i α}{p^{2}}} \dots \dots . e^{\frac{i α}{p}}, p \in N (where i = \sqrt{- 1}), then find the value of |lim_{n \to \infty} f_{n} (π)|$

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$\begin{array}{l} f_{p} (α) = e^{\frac{i α}{p^{2}} (1 + 2 + \dots + p)} = e^{\frac{i α}{2} (1 + \frac{1}{p})} \\ ∴ lim_{n \to \infty} f_{n} (π) = lim_{n \to \infty} e^{\frac{i π}{2} (1 + \frac{1}{n})} = e^{\frac{i π}{2}} \\ \Rightarrow |lim_{n \to \infty} f_{n} (π)| = |e^{\frac{i π}{2}}| = 1 \end{array}$

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Introduction to limits

Remember concepts with our Masterclasses.

Let fp(α)=eiαp2⋅e2iαp2…….eiαp, p∈N where i=−1 , then find the value of limn→∞ fn(π)

fp(α)=eiαp2(1+2+…+p)=eiα21+1p∴limn→∞ fn(π)=limn→∞ eiπ21+1n=eiπ2⇒limn→∞ fn(π)=eiπ2=1

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$Let f_{p} (α) = e^{\frac{i α}{p^{2}}} \cdot e^{\frac{2 i α}{p^{2}}} \dots \dots . e^{\frac{i α}{p}}, p \in N (where i = \sqrt{- 1}), then find the value of |lim_{n \to \infty} f_{n} (π)|$