FURTHER N-TERM GENERALIZATION OF DECIMAL POINT SERIES

 
 
 
 
In the 3-term case, where (starting with 0, 0, 1) we successively combine the last 3 terms to obtain the next term, the relevant series is 1, 1, 2, 4, 7, 13, 24, 44, 81, 149,

Therefore the corresponding decimal point series is

.01 + .001 + .0002 + .00004 + .000007 + .0000013 + .00000024 + .000000044 + .0000000081 +

As we have seen the sum of this series = 1/88.9

So we see that in the three-term case the result =1/ 88.9,

Here 88.9 = 100 - 10 - 1 - .1.

This result can be generalized for cases where multiple combinations of the three terms are combined

When (starting with 0, 0, 1) when we combine twice the last term, with the previous two terms (i.e. a = 2, b= 1, c = 1),

we get

0, 0, 1, 2, 5, 13, 33, 84, 214, 545, 1388,

Here the result of the corresponding decimal point series is related to the reciprocal of

100 - 2(10) - 1 - .1 = 78.9.

Thus 1/78.9 = .01 + .002 + .0005 +.00013 + .000033 + .0000084 +

When we combine the last with twice the second last and (once) the previous term (i.e. a = 1, b= 2, c = 1), we get

0, 0, 1, 1, 3, 6, 13, 28, 60, 129, 277, 595,

Here the result of the corresponding decimal point series is related to the reciprocal of

100 - 10 - 2(1) - .1 = 87.9.

Thus 1/87.9 = .01 + .001 +.0003 + .00006 + .000013 + .0000028 + .0000060 + ..

When we combine the last with the second last and twice the previous term (a = 1, b = 1, c = 2), we get

0, 0, 1, 1, 2, 5, 9, 18, 37, 73, 146, 293,.

Here the result of the corresponding decimal point series is related to the reciprocal of

100 - 10 - 1 - 2(.1) = 88.8.

Thus 1/88.8 = .01 + .001 + .0002 + .00005 + .000009 + .0000018 + .00000037 +.
 
 

In general in the 3-term series when we combine a times the last term with b times the second last and c times the third last terms the resultant decimal point series - the reciprocal of 100 - 10a - b - .1c (where a , b and c < 9).

Thus when a = 2, b = 1, c = 3,

we get

0, 0, 1, 2, 5, 15, 41, 112, 310, 855, .

Thus the sum of the corresponding point series = the reciprocal of 100 - 20 - 1 - .3 = 78.7

Thus 1/78.7 = .01 + .002 + .0005 + .00015 + .000041 + .0000112 + .00000310 + .000000855 +
 
 
 
 

Thus in the 4-term series (starting with 0, 0, 0, 1) the sum of terms of the relevant decimal point series = the reciprocal of 100 - 10a - b - .1c - .01d (where a, b, c, and d < 9).

These results can be generalized to n terms.

Thus in the n-term series starting with 01, 02, 03, .0n - 1, 1) the sum of terms of the relevant point series = the reciprocal of 100 - 10a - b - .1c - . . 0102,03,0n - 31p (where a, b, c, . p < 9)
 
 

Likewise in the n-term case where we alternate the signs of the decimal point terms,

the sum of terms of relevant point series = the reciprocal of 100 + 10a - b + .1c - . . 0102,03,0n - 31p (where a, b, c, . p < 9).
 
 

These results can of course be extended as before to bases other than 10.