KVPY SX SB Exam 19-Nov-2017 Question Paper

Question : 87 of 120

Marks: +1, -0

Let

a_n = ∫↖{π}↙{-π}{| x -1|}\cos nx dx

for all natural numbers n. Then the sequence

(a_n)_{n≥0}

satisfies

Solution:

a_n ={ ∫↖{1}↙{-π} {-( x -1)}\cos nx dx }↙{\I}

+ {∫↖{π}↙{1}{( x -1 )}\cos nx dx}↙{\II}

Solve I & II part by I LATE

a_n = 2π\sin{nπ}/n+2/n^2\cosnπ=2/n^2\cosn

\lim↙{n→∞} a _n=0

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