KVPY SB SX Exam 2011 Question Paper

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Question : 88 of 120

Marks: +1, -0

For each positive integer n, define

f_n(x)=\minimum (x^n/n!,(1-x)^n/n!)

, for

0≤ x ≤1.

Let

I_n=∫↖{1}↙{0}f_n(x)dx,n ≥ 1.

Then

∑↖{∞}↙{n=1}I_n

is equal to

$2√e-3$
$2√e-2$
$2√e-1$
$2√e$

Solution:

I_n=∫↖{1\/2}↙{0}x^n/n!dx+∫↖{1}↙{1\/2}(1-x)^n/n!dx

=1/(n+1)!((1/2)^{n+1}+(1/2)^{n+1})

=(1/2)^n/(n+1)!

∑↖{∞}↙{n=1}I_n=({1\/2}/2!+({1\/2})^2/3!+...)

=2√e-3

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