Htykuut

I can not find what is an infinite sequence of this type, where the coefficient for each next term is 1 more than the previous one. Help me please. I think it's interesting.

NESTED SQUARE ROOTS SERIES

Hi Stephan. Yes, I found your problem interesting and I think I have a solution.

The iterator value "i" is put as a mutiplier of the "a" of the root series. To start with the maximum value of the iterator put in coefficient of the a in the 1st iteration and end with 1 as "a" coef in the last iteration, we'll have a make a decreasing iteration.
Starting with t=0 allows a clean initial value (change and see...).

In[1]:= For[i = 5; t = 0, i >= 1, i -= 1, t = (t + i a)^(1/2)]; t

Out[1]= Sqrt[a + Sqrt[2 a + Sqrt[3 a + Sqrt[Sqrt[5] Sqrt[a] + 4 a]]]]

Of course at the end of the above displayed result Sqrt[Sqrt[5] Sqrt[a]+4 a] should read Sqrt[4 a +Sqrt[5 a] ], which is in the desired form.

Let' s see what this gives with a=4 and 5 iterations:

In[2]:= For[i = 5; t = 0, i >= 1, i -= 1, t = (t + i 4)^(1/2)]; t
N[t]

Out[2]= Sqrt[4 + Sqrt[8 + Sqrt[12 + Sqrt[16 + 2 Sqrt[5]]]]]

Out[3]= 2.73377

With the same a=4 and 15 iterations the output remains very close. The series is rapidly convergent.

In[4]:= For[i = 15; t = 0, i >= 1, i -= 1, t = (t + i 4)^(1/2)]; N[t]

Out[4]= 2.73397

Let's build a summarized functions[a_,tini_,imax] where we enter the "a" value, the start value tini of the function, and the maximum number of iterations imax:

In[5]:= s[a_, tini_, imax_] :=
For[i = imax; t = tini, i >= 1, i -= 1, t = (t + i a)^(1/2)]

Let's see the data for a=5, 0 at start, and 10 iterations, so s[6,0,10]
To show the result we have, optionally for check to ask the nested final value "t", and its approx value N[t]

In[6]:= s[6, 0, 10]
t
N[t]

Out[7]= Sqrt[6 + Sqrt[
12 + Sqrt[18 + Sqrt[
24 + Sqrt[30 + Sqrt[36 + Sqrt[42 + Sqrt[48 + Sqrt[54 + 2 Sqrt[15]]]]]]]]]]

Out[8]= 3.17874

We see that 10 more iterations do not change the first 5 decimal digits.

In[9]:= s[6, 0, 20]
N[t]

Out[10]= 3.17874

Voila!
Olivier

PS: I don`t know how to past with MTM output format, neither how to attach a notebook (it's my 1st post...). But you Will easily copy and past the formulas in a new notebook.