The expression: 3/4 + 5/36 + 7/144 + ..... + 17/5184 + 19/8100 is equal to:

Concept: For consecutive positive integers k and k+1, the fraction (2k+1)/[k2(k+1)2] simplifies exactly to 1/k2 - 1/(k+1)2 -- the difference of the reciprocals of their squares. Summing such differences for consecutive values of k telescopes: every interior term cancels, leaving only the very first and the very last term.

Application: Matching each term of the given series to this pattern:

1. 3/4 = 1/12 - 1/22 (k = 1)

2. 5/36 = 1/22 - 1/32 (k = 2)

3. 7/144 = 1/32 - 1/42 (k = 3)

4. ... the same pattern continues for k = 4, 5, 6, 7

5. 17/5184 = 1/82 - 1/92 (k = 8)

6. 19/8100 = 1/92 - 1/102 (k = 9)

7. Adding all nine differences telescopes: every middle term (1/22, 1/32, ..., 1/92) appears once with a plus sign and once with a minus sign and cancels out, leaving only 1/12 - 1/102.

8. 1/12 - 1/102 = 1 - 1/100 = 99/100 = 0.99

Cross-check: Add just the first two terms directly: 3/4 + 5/36 = 27/36 + 5/36 = 32/36 = 8/9, which equals 1/12 - 1/32 = 1 - 1/9 = 8/9. The partial telescoping matches independently, confirming the pattern used for the full nine-term sum.

So the expression equals 0.99.