In a real analysis exam last year, I had to show if the following sequence $(u_n)_{n \in \mathbb{N}}$ defined by $u_n = \frac{2^n}{n + 2^n}$ is summable using property of sequences. However, I failed to justify correctly that it was not summable.
I tried to squeeze the sequence between $0$ and $1$ :
$$ 0 < \frac{2^n}{n + 2^n} < 1 $$
And because $(0)_{n \in \mathbb{N}}$ is summable but $(1)_{n \in \mathbb{N}}$ is not summable, I concluded that $\left(\frac{2^n}{n + 2^n}\right)_{n \in \mathbb{N}}$ is not summable.
I guess my squeezing is wrong. I was lately thinking about :
$$ 0 < \frac{2^n}{n + 2^n} < \frac{2^n}{n} $$
instead and I would be tempted to conclude that $\left( \frac{2^n}{n} \right)_{n \in \mathbb{N}}$ is not summable so $\left(\frac{2^n}{n + 2^n}\right)_{n \in \mathbb{N}}$ is not summable by comparison. Is it right ?