Math - Sum of Fractions

Question

Find the sum of the series 18/12+18/20+18/30+18/42+...+18/1260

Answer 1

factoring out 18/2 = 9, the sum is 1/6 + 1/10 +1/15 + 1/21 + 1/28+ 1/36 + 1/45 + 1/55 + ... + 1/630

since the difference between consecutive terms IN THE DENOMINATOR is n+1, then the

denominator has an^2 + bn+ c as the general term

f(1) = 6 ---> a+b+c = 6

f(2) = 10 ---> 4a + 2b + c = 10

f(3) = 15 ---> 9a + 3b + c = 15

3a + b = 4

5a + b = 5

2a = 1

a = 1/2 ---> b = 5/2 ---> c = 3

(1/2)n^2 + (5/2)n + 3 = f(n)

(1/2) (n^2 + 5n + 6) = f(n)

(1/2) (n+3)(n+2) = f(n) is the general term for the denominator

Each summand is (2) (1/[(n+2)(n+3)]) =

-2 [(1/(n+3))- (1/(n+2))] <-- partial fraction decomposition

= 1/(-1/2) [[(1/(n+3))- (1/(n+2))] <--- formula for the sum of the telescoping series

-2 sum [ 1 / (1 - ( (-1) - m )) ]

-2 * sum [ 1 / ( 1 + (m+1)) is the sum of the sequence

-2 * sum [ 1/(m+2)]

Please refer to the wikipedia article: "Series summation of rational function with quadratic denominator" for more info. and the algebra

Per excel, The sum is 0.30555555...... = 0.305

= 0.3 + 0.00555555.......

= 0.3 + (1/1000)(5/9)

= 3/10 + 5/9000

= (3*900 + 5)/9000

= 2705/9000

= 541/1800

the factors of 1800 = 18*100 = 3*3*2 * 5*2*5*2

541 does not divide by 3,2, nor 5 so this horrible fraction is reduced.