Interacting Integ

Calculus Level 5

Let I n = ∫ 0 π cos ⁡ n x d x 13 − 12 cos ⁡ x I_{n} = \displaystyle \int_{0}^{\pi} \dfrac{\cos n x \; \mathrm dx}{13-12\cos x} .

Compute the value of lim ⁡ n → ∞ ( I 0 + I 1 + … + I n ) \displaystyle \lim_{n\to \infty}(I_{0}+I_{1}+\ldots+I_{n})

If your answer comes in form, a π b \dfrac{a\pi}{b} , where a a and b b are coprime positive integers , then find a + b a+b .


The answer is 8.

Anish Harsha by Anish Harsha , 20, India

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1 solution

Important hint - Obtain a homogeneous linear recurrence relation in I n I_n , I ( n − 1 ) I_(n-1) and I ( n − 2 ) I_(n-2) .Find the values of I 1 I_1 and I 2 I_2 .Using characteristic polynomial , find general solution for I n I_n .You will get I n I_n = p i / 5 pi/5 t i m e s times ( 2 / 3 ) n (2/3)^n .Now, evaluate the limit using infinite GP summation.

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