The interval [0,4] is partitioned into n equal subintervals, and a number x_i is arbitrarily chosen in the i^th subinterval for each i. Then....
lim_(n → ∞) ∑_(i = 1 → n)[2x_i+8)/n]=???
12.5
To find the limit as n approaches infinity of the given expression, we can use the concept of a Riemann sum.
First, let's choose a representative point x_i in each subinterval. Since the subintervals are equal in size, we can choose the representative point to be the midpoint of each subinterval. Therefore, x_i = (i - 0.5) * (4/n), since there are n subintervals in total.
Now, let's substitute the chosen representative point into the expression: (2x_i + 8)/n.
Substituting x_i = (i - 0.5) * (4/n), we get:
(2 * (i - 0.5) * (4/n) + 8)/n.
Expanding this expression, we have:
[8(i - 0.5)/n + 8]/n.
Simplifying further, we have:
[8i/n - 4/n + 8]/n.
Next, let's find the sum of the expression over all i from 1 to n:
∑(i = 1 to n) [8i/n - 4/n + 8]/n.
Expanding the summation, we have:
(8/n) * ∑(i = 1 to n) i/n - (4/n) * ∑(i = 1 to n) 1 + (8/n^2) * ∑(i = 1 to n) 1.
Simplifying each term of the sum:
(8/n) * [1/n * ∑(i = 1 to n) i] - (4/n) * n + (8/n^2) * n.
Now, let's compute each term separately:
1/n * ∑(i = 1 to n) i is the average value of the integers from 1 to n, which is (n + 1)/2.
So, the expression becomes:
(8/n) * [(n + 1)/2] - (4/n) * n + (8/n^2) * n.
Simplifying further, we have:
4(n + 1)/n - 4 + 8/n.
Taking the limit as n approaches infinity:
lim_(n → ∞) [4(n + 1)/n - 4 + 8/n].
Now, let's analyze each term as n approaches infinity:
- As n becomes very large, (n + 1)/n approaches 1.
- As n becomes very large, 8/n approaches 0.
Therefore, the expression becomes:
4(1) - 4 + 0 = 4 - 4 + 0 = 0.
Hence, the limit as n approaches infinity of the given expression is 0.