4- 1 - 24 Harmonic Series Proof : As always, let us define s,= 2 n We are going to focus on powers of ? ( partial sun Let us start with So . Sq=Itz Sy =, Itz + 3 + ( 9 2 1 + 3 + 9 + 9 = 1+ 2 +5 = 1+3 V = 1+2 71+ 5+ 3+5+2 = 1+2 S,2zltz But how much is the limit 120 1+2 - 00 . Therefore : 900 so: 200aday since all the terms I am summing are positive, y Si is always increasing , Therefore isas S; = 00Question : Timit How does isoo Sai zoo relate to 2 me? 0 I n = : 200 S. = 1 300 Tim n=1 2 2 22 23 24 -play Since all the terms I am summing are positive, y s; is always increasing Therefore lim 130 s; = 00