Proving Divergence of (1+1)/(1+2) + (1+2)/(1+4) + ... + (1+n)/(1+2n) Series

[Jamin2112]

Homework Statement

Explain why the series

(1+1)/(1+2) + (1+2)/(1+4) + ... + (1+n)/(1+2n) + ...

is divergent.

Homework Equations

For a series to ∑u_n to be convergent, it is necessary that lim _n-->∞ u_n = 0.

The Attempt at a Solution

As you may have guessed, I'm going to show that lim _n-->∞ (1+2n)/(1+n) ≠ 0.

Assume lim _n-->∞ (1+2n)/(1+n) = 0. Let ∂ > 0, and then there exists an integer N such that

|(1+2n)/(1+n)| = (1+2n)/(1+n) < ∂

whenever n ≥ N.

(1+2n)/(1+n) < ∂ ----> (1+n)/(1+2n) > 1/∂ ----> 1 - n/(1+2n) > 1/∂ ----> n/(1+2n) > 1/∂ - 1.

Hmmmm ... Now how do I explain that n/(1+2n) goes to zero, and thus will eventually be smaller than 1/∂ - 1?

 

[Office_Shredder]

First, you're looking at the reciprocal of the term you're interested in. And when you split your fraction up you lost the denominator in one of the parts

Try dividing the numerator and denominator by n at the start

 

[Jamin2112]

First, you're looking at the reciprocal of the term you're interested in. And when you split your fraction up you lost the denominator in one of the parts

Try dividing the numerator and denominator by n at the start

Of course I know that (1+2n)/(1+n) = (1/n + 2)/(1/n + 1), and that the limit obviously equals 2.

I was just trying to do it with the definition of a limit.