Question

If [x] denotes the greatest integer function less than or equal to x, then [ ( 1 + 0.0001)¹⁰⁰⁰⁰] equals to?

Answer 1

It is well known that lim n—> inf (1 + 1/n)^n evaluates to e.
So if we put n = 10000
(1 + 1/10000)^10000
And by binomial theorem,

Using the binomial theorem:
( 1 + 1/10000 )^10000 = 10000C0 + 10000C1*(1/10000) + 10000C2 * (1/10000)^2 + 10000C3*(1/10000)^3 + ……….. + 10000C10000 * (1/10000)*10000
= 1 + 1 + 10000C2*(1/10000)^2 + 10000C3*(1/10000^3) + …….
Now, 10000C2 = 10000*9999/2 , which is obviously less than 10000*10000/2 = (1/2)*1000^2
Similarly , 10000C3 = 10000*9999*9998 / 6 , which is less than 10000*10000*10000 / 6 = (1/6)*10000^3
and so on….
so (1 + .0001)^10000 = 1 + 1 + 10000C2*(1/10000^2) + 10000C3*(1/10000^3 ) + ……. + 10000C10000*(1/10000^10000)
So, (1 + .0001)^10000
==> (1 + .0001)^10000
Now, we know that 2 + 1/2 + 1/4 + 1/8 + …. upto infinity =3
==> (1 + .0001)^10000
2
=> [x] = 2












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