Doubt with another form of Balmer' Series In the book I'm studying from (Solid state electronic devices - Ben Streetman) it is said that the Balmer series is defined as follows:
$$
v = cR \left( \frac{1}{2^2} - \frac{1}{n^2}\right)
$$
with c speed of light and $R = 109,678 [cm]^{-1}$. But in the majority of other books and sites we can find the series as
$$
v = R_H \left( \frac{1}{2^2} - \frac{1}{n^2}\right)
$$
with $R_H = 10973731.57 m^{−1}$ or, in its wavelength form:
$$
\lambda = b\left ( \frac{n^2}{n^2-4} \right)
$$
with $b = 364.56 [nm]$
So, the problem is that I can't figure out a direct relation between the equation from Streetman's book and the latter. Basically if we work it out we should get that $\frac{4}{cR} = b$ or $\frac{1}{cR} = R_H$ but using $c = 2.998*10^{10} [cm/s]$ and $R = 109,678 (cm)^{-1}$ clearly what I said doesn't hold. 
 A: The answer depends a little on history.
Firstly, the Bohr's model of the atom can be used to derive the theoretical prediction of the hydrogen atom transition wavelenghts as:
$$\frac{1}{\lambda} = R\left(\frac{1}{n_L^2} -  \frac{1}{n_U^2}\right)$$
In this case, $\lambda$ is the wavelength, and $R$ is the Rydberg constant defined as:
$$R=\frac{me^4}{8 \epsilon_0^2 h^3 c}$$
If you substitute the numerical values for $m, c, e, h$ and $\epsilon_0$, you will get a value of $R \approx 10973731.5m^{-1}$. Note $c$ is the speed of light and $h$ is Planck constant.
Now, in Johann Balmer's time, the Bohr's model of the hydrogen atom was not available. Balmer's equation was 
$$\frac{1}{\lambda} = R_B \left(\frac{1}{2^2} -  \frac{1}{n^2}\right)$$ where $R_B$ (due to Balmer) was experimentally determined (in order to fit the spectral observations) as: $R_B=109678cm^{-1}=10967800m^{-1}$.
Note that $\frac{R}{R_B} \approx 1.00054$.
Now, notice that if we denote $\nu$ as the frequency in Hertz, then:
$$\frac{1}{\lambda} = \frac{\nu}{c}= R \left(\frac{1}{2^2} - \frac{1}{n^2}\right) \Rightarrow \nu = cR \left(\frac{1}{2^2} - \frac{1}{n^2}\right) \approx cR_B \left(\frac{1}{2^2} -  \frac{1}{n^2}\right)$$
where $c$ is the speed of light.
Summary: In your first equation $\nu$ refers to the frequency of the atomic transition. In the second equation $\nu$ is defined as $\nu=\frac{1}{\lambda}$.
Also note the difference between the reciprocal of the wavelength, $\frac{1}{\lambda}$, and the wavenumber, $|\mathbf{k}|$, which is typically defined as $\frac{2\pi}{\lambda}$.
