Prove that the sequence {Xn} is a divergent sequence where Xn= 1+ 1/2+ 1/3+ 1/4+.....+1/n for any natural numbers n.

We need to prove that the series Xn is divergent where xn=1+1/2+1/3+1/4+....+1/n

To prove that we need to come up with another series its values equals or smaller than xn. This is called the Comparison test.

Now. let's assume the series 1+1/2+1/2+1/4+1/4+1/8+1/8+1/8+1/8+1/16+....= sum yn

Now comapre it to the seires  xn

then,

1+1/2+ 1/3+1/4 + 1/5+ 1/6+1/7+1/8+ 1/9+.....+1/n = sum xn

1+1/2+(1/4+1/4)+(1/8+1/8+1/8+1/8)+(1/16....)  = sum yn

1+1/2+1/2+1/2+1/2..... = sum yn = inf

But we observe that xn values equal or greater than yn values. Since sum yn --> inf, then sum xn-->inf

Then Xn is divergent.