Find a symmetric random walk on $\mathbb{Z}$ that is transient. I wanted to know if it is possible find a symmetric random walk on $Z$ that is not recurrent. Let $X$ have the following distribution, with a probability $1/2^{i+1}$, $X=\pm b_i$. 
Let $$S_n=\sum_{k=1}^n X_k$$
be the random walk on $\mathbb{Z}$ with $X_k$ having the same distribution as $X$. The problem is to find $b_n$ such that $S_n$ is transient.
We need to find $b_n$ large enough so that $E(X)$ is not defined (otherwise it would be 0 by symmetry and thus recurrent). I think $b_n=2^n$ is sufficient, but am having trouble showing transience. 
We know that a random walk is transient iff $$\int_{-\pi}^\pi \frac{1}{1-\varphi_X(t)}dt<\infty$$
I computed the characteristic function of $X$ as follows:
$$\varphi_X(t)=\sum_{n=1}^\infty \frac{1}{2^{n+1}}e^{it2^n}+\frac{1}{2^{n+1}}e^{it2^{-n}}=\sum_{n=1}^\infty \frac{\cos(t2^n)}{2^n}$$
But I am unsure of how to proceed.

*

*$\varphi_X(t)$ is periodic (with period $\pi/2$). So, we may consider a small neighborhood around zero, say $(-\delta,\delta)$

*As I mentioned in the comments $b_n=2^n$ is not enough to get transience. So, let $b_n=4^n$

*For $t\in(-\delta,\delta)$ and some constant $c$
$$1-\varphi_X(t)=\sum_{n=1}^\infty \frac{1-\cos(t4^n)}{2^n}\ge c\sqrt{|t|}$$


*Finally,


$$\int_{-\delta}^\delta \frac{1}{1-\varphi_X(t)}dt\le \int_{-\delta}^\delta \frac{1}{c\sqrt{|t|}}dt<\infty$$
So, the random walk is transient.
