Integration at its Best!!!!!!!!

Calculus Level 5

$\lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ \infty }{ \cos { { x }^{ n } } dx } }$

0 e

e^{e}

\infty

1 -

\infty

1 solution

Rajdeep Dhingra
Feb 3, 2015

$\lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ \infty }{ \cos { { x }^{ n } } dx } } \\ =\quad \lim _{ n\rightarrow \infty }{ \Gamma (1+\frac { 1 }{ n } )\cos { \frac { \pi }{ 2n } } } \\ =\quad \Gamma (1)cos0\\ =\quad \boxed { 1 }$

Nice. What was my fault of converting into

$^\lim_{n \to \infty} \int_0^{\infty} \dfrac{cos x d x}{n x} = 0?$ I think the step of conversion is correct.

I forgot $^\lim_{n \to \infty} \int_0^{\infty} \dfrac{cos x d x}{x} =$ not converged.

n tends to infinity, Integrate from 0 to infinity of (cos x/ (n x)) = 0 by WolframAlpha but saying "Standard computation time exceeded..."

I didn't see 1 as one of them in a situation of uncertain. Not the first time I made similar mistake.

Lu Chee Ket - 5 years, 5 months ago

Integration at its Best!!!!!!!!

1 solution

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