Question

show that 1-1/2+1/3+1/4+...+1/99-1/100<4/5
show that​

Answer 1

You can draw the graphs of y=1x and y=1x+1 and see that

1+∫1001dxx+1<∑100n=11n<1+∫1001dxx.

Evaluating the integrals yield

1+ln(101)−ln(2)<∑100n=11n<1+ln(100).

OK, 1+ln(100)≈5.6 , but 1+ln(101)−ln(2)≈4.92 . So we need to refine this estimate. We can begin the approximation later:

1+12+∫1002dxx+1<∑100n=11n

and this works well, since the integral evaluates to 1+12+ln(101)−ln3≈5.01 . Then 5<∑100n=11n<6 , and the integer part of the desired sum is 5.