解：
$\begin {aligned}&2×(1+\frac {1}{2})×(1+\frac {1}{2^2})×(1+\frac {1}{2^4})×(1+\frac {1}{2^8})+\frac {1}{2^{14}}\\=&4×(1-\frac {1}{2})×(1+\frac {1}{2})×(1+\frac {1}{2^2})×(1+\frac {1}{2^4})×(1+\frac {1}{2^8})+\frac {1}{2^{14}}\\=&4×(1-\frac {1}{2^2})×(1+\frac {1}{2^2})×(1+\frac {1}{2^4})×(1+\frac {1}{2^8})+\frac {1}{2^{14}}\\=&4×(1-\frac {1}{2^4})×(1+\frac {1}{2^4})×(1+\frac {1}{2^8})+\frac {1}{2^{14}}\\=&4×(1-\frac {1}{2^8})×(1+\frac {1}{2^8})+\frac {1}{2^{14}}\\=&4×(1-\frac {1}{2^{16}})+\frac {1}{2^{14}}\\=&4-\frac {1}{2^{14}}+\frac {1}{2^{14}}\\=&4\end {aligned}$
