Upper bound of $\left(1+\dfrac{1}{2}\right)\left(1+\dfrac{1}{4}\right)...\left(1+\dfrac{1}{2^n}\right)$

Question

In a calculus book that I'm reading, there is a problem as follows:

Prove that this sequence $x_n=\left(1+\dfrac{1}{2}\right)\left(1+\dfrac{1}{4}\right)...\left(1+\dfrac{1}{2^n}\right)$ is monotonic, bounded and then converges.

I can prove that this sequence is increasing, but I found no way to find the upper bound for $x_n$.

Please give me some hint to find the upper bound, and may be the limit of this sequence also.

Thanks

Answer 1

Let,

$$u_n = \prod_{i=1}^n \left( 1+ \dfrac{1}{2^i}\right)$$

Hint:

Calculate $\ln(u_n)$ and use that $\ln(1+x) \leq x$ for all $x$.

Once you've shown that $(u_n)_n$ is bounded, as $(u_n)_n$ is an increasing sequence it follows that $u_n$ converges by the monotone convergence theorem.

Answer 2

Using GM-AM you get

$$\prod_{k=1}^n \left(1+\frac 1{2^k}\right)< \left(\frac{n+1}{n}\right)^n < e$$