The series ∑uᵣ (r = 1 to r = N) is essentially the sum of the sequence up to N terms.


SN = (3N² - 5) / N(N+1)

= (3N² - 5) / (N²+N)

= (3N² + 3N - 3N - 5) / (N²+N)

= 3 - (3N + 5) / (N²+N)

= 3 - (3N + 5) / N(N+1)



As N increases and approaches infinity, N(N+1) is increasing at a much faster rate than 3N + 5 .

(linear expression with positive coefficient of N and constant for numerator,

vs.

quadratic expression with positive coefficient of N² and N in the denominator. It is also equal to N(N+1) so as N increases, N+1 increases too and their product will be much bigger than 3N + 5 , where N is multiplied by a constant

Both are always positive since N ≥ 1 but quadratic one increases faster

This means that the difference between them (numerator and denominator) is getting wider.


You can try plotting the graphs of y = 3x + 5 and y = x² + x on a GC or Desmos and you will see that 3x + 5 is bigger than x² + x up to x ≈ 3.449. After that, x² + x overtakes and the gap between them gets increasingly wider.





So the fraction (3N - 5) / N(N+1) is decreasing as N increases , getting smaller and smaller and → 0

The series coverges to 3 and sl the sum of infinity is 3.

2 × 10-² = 2/10² = 2/100 = 0.02

When SN is within 0.02 of the sum to infinity, it is smaller than it by up to 0.02, but not inclusive.

i.e less than 0.02 below S∞

SN cannot be bigger than 3 since it is increases from a smaller value and converges to 3.


0 < S∞ - SN < 0.02

0 < 3 - (3N²-5)/(N²+N) < 0.02

0 < 3 - (3 - (3N+5)/(N²+N)) < 0.02

0 < (3N+5)/(N²+N) < 0.02


Using GC,

when (3N+5)/(N²+N) = 0.02,

N ≈ -1.6594 (reject as N ≥ 1)

or N ≈ 150.659

When (3N+5)/(N²+N) = 0,

3N + 5 = 0 since denominator cannot be 0.

N = -5/3 (reject as well)



Since N is an integer, smallest N = 151