Chapter 31 - Atomic Physics - Conceptual Questions - Page 1111: 6

Please see the work below.

We know that $\frac{1}{\lambda}=R(\frac{1}{n^{\prime}2}-\frac{1}{n^2})$ We also know that $n=n^{\prime}+1, n^{\prime}+2,.......... \infty$ For $n=1$, we obtain: $n=2,3,4,....... \infty$ Thus, if n tends to $\infty$, then we have an infinite number of spectral lines for a given series. But when 'n' keeps on increasing, we obtain more closely packed spectral liens which are hard to distinguish.