Question

What do you think about it? How to prove it? or it's not true

1/() `1/(n+1) + 1/(n+2)+ cdots+1/(3n+1) >1`

Answer 1

See below.

Explanation:

Assuming that the question is about

`S_n = (sum_(k=1)^(2n+1) 1/(n+k)) > 1` we will demonstrate it using finite induction.

1) `S_1 = 1/2+1/3+1/4 = 13/12 > 1`

2) Now assuming that `S_n = (sum_(k=1)^(2n+1)1/(n+k)) > 1` we have

3) `S_(n+1)=sum_(k=1)^(2(n+1)+1)1/(n+1+k) = S_n - 1/(n+1)+1/(3n+2)+1/(3n+3)+1/(3n+4) > 1`

And thus we can conclude that

`S_n = (sum_(k=1)^(2n+1)1/(n+k)) > 1, forall NN^+`

NOTE

`1/(3n+2)+1/(3n+3)+1/(3n+4)-1/(n+1) = 2/(3 (1 + n) (2 + 3 n) (4 + 3 n)) > 0`

`lim_(n->oo)S_n = log_e 3`