Trying again to understand Piketty

Earlier mistake (repeated here) was to add the whole gain from capital to deduce the next year's capital. We use a fraction of it in the next calculations.
β is C/I where C is capital and I is income. To consider the long term changes, I will denote these quantities at a given time by subscripts β_t etc and after an year by β_(t+1) and growth rate for that year by g_t etc. f_t is the fraction of the return from capital that is saved and added to the capital for the next year.

So C_(t+1)=C_t+f_t*r_t*C_t and I_(t+1)=I_t+g_t*I_t (here * denotes multiplication which I will omit it later and just use juxtaposition); If the population is not stagnant, growth rate in the population is added in g_t. Dividing both sides by I_(t+1) and simplifying, we have
 β_(t+1)/β_t=(1+f_t r_t)/(1+g_t) (call this Equation A) 
(Notice that if the national savings rate is s_t, we should have s_t =f_t r_t β_t (Equation A'))
from which we can estimate β_(t+k)/β_0 for k years as  a product of successive ratios:
β_(t+k)/β_0=((1+f(t+k-1)r(t+k-1)/(1+g_(t+k-1)).....((1+f_0 r_0)/(1+g_0)) (Equation  B)

By rearranging we also get another identity comparing f_t, r_t , g_t:

f_t r_t-g_t=((β_(t+1)/β_t)-1)(1+g_t) (call this Equation C)

Piketty's observation is that this difference r-g has been generally positive except for a  period 1913-1970 in the twentieth century and often f_t r_t is greater than g_t. Generally β has been less than 8 or so. Assuming that growth rate is bigger than -1 during  the period under consideration, we see from Equation C that for f_t r_t-g_t to be positive we also should have 
(β_(t+1)/β_(t)-1)greater than 0 or β_(t+1) greater than β_t. That is, β_t increases during that period. If it does not go to infinity, we should have β_t converging to some β in the long run during the period when f_t r_t-g_t greater than 0. This convergence implies from Equation A that 1+f_t r_t gets close to 1+g_t or that f_t r_t and g_t are nearly equal after a while. If we assume that g_t converges to some g, then f_t r_t also converges to the same number g.

So the conclusion is that if β_t does not explode, whatever the growth rate is (as long as it is bigger than -1) f_t r_t modifies to mimic the behaviour of g_t.

From equation A', we see that β_t converges to the limit of s_t/f_t r_t =s/g which corresponds to the asymptotic formula β=s/g.

Of course, I am not sure whether this is correct either. But these are my models for understanding parts of Piketty's Capital.

P.S. Calculations of this type are in a paper of Thomas Piketty and Gabriel Zucman  http://behl.berkeley.edu/files/2013/02/Piketty-Zucman_WP2013-10.pdf