Let n be the smallest positive integer such that 1+ 1/2+1/3+....+1/n\geq4 . Which one of the following statements is true ?

Correct Answer is : 20 < n ≤ 60

KVPY SX SB Exam 19-Nov-2017 Question Paper

Question : 9 of 120

Marks: +1, -0

Let n be the smallest positive integer such that

1+ 1/2+1/3+....+1/n≥4

.
Which one of the following statements is true ?

Solution:

1 + x < 1 + x + x^2/2!+x^3/3!+...

1+ x < e^x

(1+x) < x

x=1/y

(1+1/y)<1/y

(y + 1) -\ln y <1/y

put y = 1
ln (2) – ln (1) <

1/1

put = y = 2
ln 3 – ln 2 <

1/2

put = y = 3
ln 4 – ln 3 <

1/3

put y = n
ln (n +1) -ln n <

1/4

ln (n +1) – ln (1) <

1+1/2+1/3+... 1/n

ln (n +1) ≤ 4

n ≤ e^4-1

n ≤ 60

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