May 11th, 2019, 03:19 PM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 510 Thanks: 79  Integer part
Compute the integer part of $\displaystyle y=\sqrt{1} +\sqrt{2} +... +\sqrt{12}$.(without calculator) $\displaystyle \lfloor y \rfloor =$? Last edited by idontknow; May 11th, 2019 at 03:25 PM. 
May 12th, 2019, 01:54 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,755 Thanks: 695 
You need to get the square roots to enough places (2?) and then sum. If you are not allowed to use a calculator, there are at least 2 methods to get the square roots to a few decimal places.

May 12th, 2019, 03:09 PM  #3 
Senior Member Joined: Aug 2012 Posts: 2,306 Thanks: 706  There are better ways arising from various formulas for the sum of the square roots of the first n positive integers. One such can easily be derived by integrating $\sqrt x$.

May 12th, 2019, 04:04 PM  #4 
Senior Member Joined: Dec 2015 From: somewhere Posts: 510 Thanks: 79 
$\displaystyle \lfloor y \rfloor > 1+ \int_{1}^{12} \sqrt{x} dx \approx 28 \; \Rightarrow \lfloor y \rfloor =1+28=29$. Also by AMGM : $\displaystyle \lfloor y \rfloor =2+ \lfloor 12\cdot (12!)^{1/24} \rfloor $. Last edited by idontknow; May 12th, 2019 at 04:12 PM. 
May 12th, 2019, 07:07 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,623 Thanks: 2076 
Why the "1 +" and "2 +"?

May 12th, 2019, 09:29 PM  #6  
Senior Member Joined: Aug 2012 Posts: 2,306 Thanks: 706  Quote:
ps  Couple more thoughts. One, I am not sure where you get your 1 plus the integral. Two, I think it should be the integral from 1 to 13. When I worked it out by hand I got 31. No guarantees. Three, using the fact that $\sqrt x$ is monotone, there's a formula to bound the error. https://en.wikipedia.org/wiki/Riemann_sum pps  Ah you are taking the right Riemann sum. So you are correct, the integral is from 1 to 12. And to answer a question that was asked, the 1 is the leftmost Riemann rectangle with base (0,1) and height $\sqrt 1 = 1$. ppps  Right Riemann sum as in rightmost point in the interval. Also right as in correct but that's not the meaning I meant. Last edited by Maschke; May 12th, 2019 at 10:10 PM.  
May 12th, 2019, 10:01 PM  #7  
Senior Member Joined: Dec 2015 From: somewhere Posts: 510 Thanks: 79  Quote:
 
May 12th, 2019, 10:05 PM  #8 
Senior Member Joined: Dec 2015 From: somewhere Posts: 510 Thanks: 79  Example : $\displaystyle x\in \mathbb{N}$ and $\displaystyle x>27,33$. $\displaystyle x>27,33$ means $\displaystyle x>28$ or $\displaystyle x\geq 29$.(we need the upper bound of 27,33 so avoid 28 in interval of x) Simply add +1 to 28 or add +2 to 27 . Last edited by idontknow; May 12th, 2019 at 10:11 PM. 
May 12th, 2019, 10:08 PM  #9 
Senior Member Joined: Aug 2012 Posts: 2,306 Thanks: 706  
May 12th, 2019, 11:49 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 20,623 Thanks: 2076 
Try it for just 6 terms.


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