Hi again, as I was doing more questions on this, I got stuck on two more questions.
1. ∑(from n=1 to infinity) ((-1)^(n+1)/ √n )
2. ∑(from n=1 to infinity) ( ln (1+1/n) )
Well Q1, same as before I tried ratio test and it turned out weird...
For Q2,not sure if I did it right, here's what I did. I compute the partial sums:
s2= ln(1+1) + ln(1+1/2) > ln(1+1/2) + ln(1+1/2) = 2 ln(1+1/2)
and so lim for n to infinity for ln(1+1/n) = infinity
therefore the sequence of the partial sum diverges to infinity and so the sequence also diverges. Is this right?
So this satisfies the alternating series test, and is therefore convergent.
Then lets evaluate this as a partial sum, looking at the telescoping aspect.
cancel out common terms, and see that -ln(1) = 0 to get
so which clearly diverges.