How do you find the integer part of the following expresion?
Find the integer part of #1/sqrt1+1/sqrt2+1/sqrt3+....+1/sqrt1000000#
Find the integer part of
1 Answer
# 1999 #
Explanation:
We seek:
# S = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(1000000) =sum_(r=1)^1000000 \ 1/sqrt(r) #
The sum:
# S_n = sum_(r=1)^n \ 1/sqrt(r) #
Is bounded from above by:
# I_u(n) = int_0^n 1/sqrt(t) \ dt#
and bounded from below by:
# I_l(n) = int_1^n 1/sqrt(t) \ dt#
Now:
# int_a^b 1/sqrt(t) \ dt = [sqrt(t)/(1/2)]_a^b = 2(sqrt(b)-sqrt(a))#
So we can write:
# I_l(n) lt S_n lt I_u(n) #
And with
# 2(sqrt(1000000)-sqrt(1)) lt S lt 2(sqrt(1000000)-sqrt(0)) #
# :. 2(sqrt(1000000)-1) lt S lt 2(sqrt(1000000)) #
# :. 2sqrt(1000000)-2 lt S lt 2sqrt(1000000) #
# :. 2*1000-2 lt S lt 2*1000 #
# :. 2000-2 lt S lt 2000 #
# :. 1998 lt S lt 2000 #
Hence, the integer part of the sum,