- #1
Jamin2112
- 986
- 12
Homework Statement
As the title says.
Homework Equations
mentioned in solution
The Attempt at a Solution
Let Sn = {(1+1)/(1+2) , (1+2)/(1+4), (1+3)/(1+6), ...}. If ∑(1+n)/(1+2n) is convergent, then lim n-->∞ Sn = 0; to put it another way, there exists an N so that whenever n ≤ N,
|(1+n)/(1+2n)|=(1+n)/(1+2n) < ∂ for all ∂ > 0.
(1+n)/(1+2n) < ∂ ----> (1+2n)/(1+n) > 1/∂ ----> 1 + n/(n+1) > 1/∂.
But since n/(n+1) < 1 for all n, the inequality 1 + n/(n+1) < 1/∂ fails when ∂ ≤ 1/2.
Thus the series converges.