For any positive integer n, let f(n) = 4n/(sqrt(2n - 1) + sqrt(2n + 1)). find the sum of f(1)+f(2)+...+ f(40)
+8 To find the sum of f(1) + f(2) + ... + f(40), we can simplify each term and then calculate the sum.
Let's simplify f(n) for any positive integer n:
f(n) = 4n / (sqrt(2n - 1) + sqrt(2n + 1))
To simplify the expression, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is (sqrt(2n - 1) - sqrt(2n + 1)):
f(n) = (4n / (sqrt(2n - 1) + sqrt(2n + 1))) * ((sqrt(2n - 1) - sqrt(2n + 1)) / (sqrt(2n - 1) - sqrt(2n + 1)))
Simplifying further, we get:
f(n) = (4n * (sqrt(2n - 1) - sqrt(2n + 1))) / ((2n - 1) - (2n + 1))
= (4n * (sqrt(2n - 1) - sqrt(2n + 1))) / (-2)
Now we can calculate the sum of f(1) + f(2) + ... + f(40):
sum = f(1) + f(2) + ... + f(40)
= (41(sqrt(21 - 1) - sqrt(21 + 1)))/(-2) + (42(sqrt(22 - 1) - sqrt(22 + 1)))/(-2) + ... + (440(sqrt(240 - 1) - sqrt(240 + 1)))/(-2)
= -2 * (1*(sqrt(21 - 1) - sqrt(21 + 1)) + 2*(sqrt(22 - 1) - sqrt(22 + 1)) + ... + 40*(sqrt(240 - 1) - sqrt(240 + 1)))
Now we can simplify each term in the parentheses and calculate the sum:
sum = -2 * (1*(sqrt(21 - 1) - sqrt(21 + 1)) + 2*(sqrt(22 - 1) - sqrt(22 + 1)) + ... + 40*(sqrt(240 - 1) - sqrt(240 + 1)))
= -2 * (1*(sqrt(1) - sqrt(3)) + 2*(sqrt(3) - sqrt(5)) + ... + 40*(sqrt(79) - sqrt(81)))
We can see that the terms inside the parentheses form a telescoping series, where the square root terms cancel each other out. The only terms that remain are the first and the last: YourTexasBenefits Login
sum = -2 * (sqrt(1) - sqrt(3) + sqrt(3) - sqrt(5) + ... + sqrt(79) - sqrt(81))
= -2 * (sqrt(1) - sqrt(81))
= -2 * (1 - 9)
= -2 * (-8)
= 16
Therefore, the sum of f(1) + f(2) + ... + f(40) is 16.
a=listfor(n, 1, 40, (2* n* (sqrt(2* n + 1) - sqrt(2* n - 1));print a, sum a
(1.464101615, 2.01606868, 2.458100001, 2.833989511, 3.166247904, 3.467117821, 3.74404899, 4.001956471, 4.244279723, 4.473535028, 4.691628224, 4.90004344, 5.09996299, 5.292346764, 5.477986671, 5.657545079, 5.831582643, 6.000578899, 6.164947788, 6.325049561, 6.481200049, 6.633677961, 6.782730723, 6.928579181, 7.071421427, 7.211435958, 7.348784282, 7.483613098, 7.616056131, 7.746235682, 7.874263952, 8.000244167, 8.124271556, 8.246434191, 8.366813718, 8.485485994, 8.602521647, 8.717986566, 8.831942334, 8.944446615)>>Sum== 242.809263
