Good Math Problem Solving Exercise: Compute the Floor of a 10,000 Terms Summation
The strategy to compute the floor is by using the fact that if we know that the value of the summation is between two consecutive integers, greater than or equal to the lower integer but less than the greater integer, then the floor of the series must be that lower integer. So we can release the floor brackets and work on the series only, by this I mean looking for the inequality:
where n is an integer. Notice that any real number have that inequality, because any number is either an integer or between two consecutive integers. To find the inequality we analyze each term of the sum, finding the narrow upper and lower bound for each term. Let’s take one term as example. 1 over square root of 2 can be written as 2 over 2 times square root of 2, by multiplying by 2/2. Do not underestimate the new denominator, we can use it to form an inequality.
As you can see, we have found a very narrow upper bound that does not involve surds, just by changing the square root of 2 with 1. You can do the same approach for all the other terms and you will get similar inequality. The minus sign suggests that we may find Telescoping pattern. The sum of the upper bound will be easier to deal with compared to the sum of the surd terms, now you just need to find an integer that bounds the sum of the upper bound. You can also use the same approach to find the lower bound of the original series.
The full solution is available on this video:
Calculate the floor of this sum without using calculator — YouTube
