Someone suggests a new way to create infinite numbers each greater than any positive constants from those Un--->0 items in Harmonic Series and turn the Un--->0 infinite Harmonic Series into a “Vn ---> any positive constants” infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000)”:
Let x:=1000000000000000 - 1. That is 10^15 - 1, which is larger than 2^49 (calculator: 2-log(10^15) is 49.82892142). The sum of x terms following 1 is
1/2+1/3+...+1/(x+1) ≧ 1/2+1/3+...+1/2^49 (which has fewer terms to add).
Now I subdivide the sequence into 49 sub-sequences T_0, T_1, T_2, .. ,T_48, where T_i runs from 1/(2^i+1) to 2/2^(i+1), thus:
T_0:= (1/2),
T_1:=(1/3,1/4),
T_2:=(1/5,1/6.1/7,1/8) etc.
It is up to you to write the intermediate 45 sequences up to T_48:=(1/(2^48+1),...1/2^49).
A generic sequence T_i has 2^i terms and sums as 1/(2^i+1) + ... + 1/(2^(i+1). Replace each term by the last one, which is the smallest:
sum(T_i) ≧ sum of (2^i times 1/2^(i+1) ) = 1/2.
The sum of all terms from 1/2 to 1/x is larger than T_0+T_1+T_2+...+T_48, which is larger that 49 times 1/2. Therefore, the desired sum is at least 24.5 (25.5, if you insist on adding the first term, 1.)
If you consider x:=1000000000000000000000000000000 =10^30 with 2-log 99.65784285, you may copy the above argument with 99 sub-sequences, summing up to at least 99/2=49.5 (50.5, if you include the first term 1).
You can already see a proof that for each positive integer x:
1+1/2+ ... + 1/(x+1) ≧ 1 + 1/2 times ( integer part of the 2-log of (x+1) ).
This matches well with a known result in calculus approximating this sum with ln(1+x).
Do you agree with it and why?
Dear Mr. Marcel Van de Vel,
Acturally, you didn’t really understand my question and have never answered my question about Harmonic Series.
My question is very clear:
1, how many numbers each greater than 1/2 do you create, do you create only one number greater than any positive constant (such as 10000000000000000000000000000000000) or many numbers each greater than any positive constant or infinite numbers each greater than any positive constant?
2, can you really turn the Un--->0 infinite Harmonic Series into a “Vn ---> any positive constants” infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000)”?
Yours,
Geng OUYANG
Five of my published papers have been up loaded onto RG to answer such questions:
1，On the Quantitative Cognitions to “Infinite Things” (I)
2，On the Quantitative Cognitions to “Infinite Things” (II)
3，On the Quantitative Cognitions to “Infinite Things” (III)
4 On the Quantitative Cognitions to “Infinite Things” (IV)
5 On the Quantitative Cognitions to “Infinite Things” (V)
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