Hello everyone!

Recently realized that ln(x) equals "1 + 1/2 + 1/3 + ... - (x-1)/x + 1/(x+1) + 1/(x+2) + ... - (x-1)/2x + + 1/(2x+1) + ...". I knew ln(2)=1 - 1/2 + 1/3 - 1/4 + ... from BpRp's videos. Can someone explain this? (I checked ln(3), ln(4) and ln(5) with calculator up to 50th term each).