- #1

- 986

- 9

## Homework Statement

As the title says.

## Homework Equations

mentioned in solution

## The Attempt at a Solution

Let S

_{n}= {(1+1)/(1+2) , (1+2)/(1+4), (1+3)/(1+6), ...}. If ∑(1+n)/(1+2n) is convergent, then lim

_{n-->∞}S

_{n}= 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.