N:=100000;evalf(sum(1/k,k=1..N));evalf(sum(1/k^2,k=1..N));evalf(sum(1/k^4,k=1..N));evalf(ln(N+1));evalf(sum(1/k^2,k=1..N)-Pi^2/6);evalf(sum(1/k^4,k=1..N)-Pi^4/90);a:=2;plot({1/x^a,[1,t,t=0..1],[2,t,t=0..1/2^a],[3,t,t=0..1/3^a],[4,t,t=0..1/4^a],[5,t,t=0..1/5^a],[6,t,t=0..1/6^a],[7,t,t=0..1/7^a],Heaviside(x)-Heaviside(x-2),(Heaviside(x-1)-Heaviside(x-3))/2^a,(Heaviside(x-2)-Heaviside(x-4))/3^a,(Heaviside(x-3)-Heaviside(x-5))/4^a,(Heaviside(x-4)-Heaviside(x-6))/5^a,(Heaviside(x-5)-Heaviside(x-7))/6^a,(Heaviside(x-6)-Heaviside(x-8))/7^a},x=0..8,y=0..2);plot({1/x^a,[3,t,t=0..1/3^a],[4,t,t=0..1/4^a],[5,t,t=0..1/5^a],[6,t,t=0..1/6^a],[7,t,t=0..1/7^a],(Heaviside(x-1)-Heaviside(x-3))/2^a,(Heaviside(x-2)-Heaviside(x-4))/3^a,(Heaviside(x-3)-Heaviside(x-5))/4^a,(Heaviside(x-4)-Heaviside(x-6))/5^a,(Heaviside(x-5)-Heaviside(x-7))/6^a,(Heaviside(x-6)-Heaviside(x-8))/7^a},x=2..8);plot({1/s^a,[1,t,t=0..1],[2,t,t=0..1/2^a],[3,t,t=0..1/3^a],[4,t,t=0..1/4^a],[5,t,t=0..1/5^a],[6,t,t=0..1/6^a],[7,t,t=0..1/7^a],Heaviside(x-1)-Heaviside(x-2),(Heaviside(x-2)-Heaviside(x-3))/2^a,(Heaviside(x-3)-Heaviside(x-4))/3^a,(Heaviside(x-4)-Heaviside(x-5))/4^a,(Heaviside(x-5)-Heaviside(x-6))/5^a,(Heaviside(x-6)-Heaviside(x-7))/6^a,(Heaviside(x-7)-Heaviside(x-8))/7^a},s=0..8,y=0..2);x:=s+1;k:='k';x:='x';int(1/x,x=1..80000);evalf(sum(1/k,k=1..79999))-int(1/x,x=1..800);for n from 0 to 6 do print(evalf(sum(1/k,k=1..8*10^n-1)-int(1/x,x=1..8*10^n))) endint(1/x,x=1..80000)-evalf(sum(1/k,k=2..80000));evalf(%);for n from 0 to 6 do print(evalf(int(1/x,x=1..8*10^n)-sum(1/k,k=2..8*10^n))) endJSFH