Question

1+1/2+1/3+1/4+...∞ == 1 is it true ​

Answer 1

Answer:

Yep, may be almost right

Answer 2

Answer:

We observe that:

1/1=1/11/1=1/1

1/1+1/2=3/21/1+1/2=3/2

1/1+1/2+1/3=11/61/1+1/2+1/3=11/6

1/1+1/2+1/3+1/4=50/241/1+1/2+1/3+1/4=50/24

1/1+1/2+1/3+1/4+1/5=274/1201/1+1/2+1/3+1/4+1/5=274/120

1/1+1/2+1/3+1/4+1/5+1/6=1764/7201/1+1/2+1/3+1/4+1/5+1/6=1764/720

1/1+1/2+1/3+1/4+1/5+1/6+1/7=13068/50401/1+1/2+1/3+1/4+1/5+1/6+1/7=13068/5040

We know that this infinite sum is equal to ζ(1)ζ(1) and it diverges to infinity.

We also observe that the numerators of the partial sums are consecutive terms of the unsigned Stirling numbers of the first kind (1,3,11,50,274,1764,13068,…)(1,3,11,50,274,1764,13068,…), while the denominators are the consecutive nn factorials (n!)(n!) for n=1,2,3,….n=1,2,3,…..

We will now prove that each numerator of the individual sums of this harmonic progression is a Stirling number of the first kind and each of them is a successive term of the sequence of the particular Stirling numbers. The proof will be done with induction.

For n=1=>1=1/1n=1=>1=1/1

For