iterate method to find roots

Question

This code gives the roots (lambda-values) of eq1 by using iteration.

eq1[n_, β_, λ_] := Hypergeometric1F1[1/4 (2 - λ/β), n + 1, β]
rootslist[n_Integer, k_Integer, β_] := 
  Rest @ FoldList[FindRoot[eq1[n, #2, λ] == 0, {λ, #1}][[1, 2]] & 
    BesselJZero[n, k]^2, Range @ β]
Table[rootslist[n, 1, 30], {n, 0, 4}]

The table contains all the λ-values for each β from 1 to 30. Using this I want to know how to find a table with $\frac{\lambda}{\beta}$ values?

For example, I have produced the following table

{6., 6.63583, 7.64905, 8.97767, 10.5495, 12.2936, 14.1498, 16.0735, 18.0349, 20.0161, 22.0073, 24.0032, 26.0014, 28.0006, 30.0003, 32.0001, 34., 36., 38., 40., 42., 44., 46., 48., 50., 52., 54., 56., 58., 60.},

for $n=0$.

How can I get

$\{\frac{6.}{1}, \frac{6.63583}{2}, \frac{7.64905}{3}, \frac{8.97767}{4}, \frac{10.5495}{5}, \frac{12.2936}{6}, \frac{14.1498}{7}, \frac{16.0735}{8},\frac{ 18.0349}{9}, \frac{20.0161}{10}, \frac{22.0073}{11}, \frac{24.0032}{12}, \frac{26.0014}{13},\frac{ 28.0006}{14}, \frac{30.0003}{15},\frac{ 32.0001}{16}, \frac{34.}{17}, \frac{36.}{18}, \frac{38.}{19}, \frac{40.}{20}, \frac{42.}{21}, ..., \frac{60.}{30}\}\text{?}$

Answer 1

Try this one:

Rational@@@Thread[{#, Range[Length[#]]}]&@rootslist[0, 1, 30]

Answer 2

I may be missing a subtlety in your question but if you just want the numeric values:

lst = {6., 6.63583, 7.64905, 8.97767, 10.5495, 12.2936, 14.1498, 16.0735, 18.0349, 
   20.0161, 22.0073, 24.0032, 26.0014, 28.0006, 30.0003, 32.0001, 34., 36., 38., 40., 42.,
    44., 46., 48., 50., 52., 54., 56., 58., 60.};

MapIndexed[#/#2[[1]] &, lst]

{6., 3.31792, 2.54968, 2.24442, 2.1099, 2.04893, 2.0214, 2.00919, 2.00388, 2.00161, 
 2.00066, 2.00027, 2.00011, 2.00004, 2.00002, 2.00001, 2., 2., 2., 2., 2., 2., 2., 2., 2., 
 2., 2., 2., 2., 2.}

If you want the visual appearance of fractions you can use Rational as Spawn did:

MapIndexed[Rational[#, #2[[1]]] &, lst]

{6./1, 6.63583/2, 7.64905/3, 8.97767/4, 10.5495/5, 12.2936/6, 14.1498/7, 16.0735/8, \
18.0349/9, 20.0161/10, 22.0073/11, 24.0032/12, 26.0014/13, 28.0006/14, 30.0003/15, \
32.0001/16, 34./17, 36./18, 38./19, 40./20, 42./21, 44./22, 46./23, 48./24, 50./25, \
52./26, 54./27, 56./28, 58./29, 60./30}

Answer 3

If what you want is just formatting your output in the described way, you can use Column/Grid.

MapIndexed[Column[{#1, #2[[1]]},
   Alignment -> Center,
   Dividers -> {False, {False, True, False}}] &,
 {6., 6.63583, 7.64905, 8.97767, 10.5495, 12.2936, 14.1498, 16.0735, 
  18.0349, 20.0161, 22.0073, 24.0032, 26.0014, 28.0006, 30.0003, 
  32.0001, 34., 36., 38., 40., 42., 44., 46., 48., 50., 52., 54., 56., 58., 60.}]