A harmonic progression is a sequence of real numbers formed by taking the reciprocals of an arithmetic progression. Equivalently, it is a sequence of real numbers such that any term in the sequence is the harmonic mean of its two neighbors.
The sequence is an arithmetic progression, so its reciprocals are a harmonic progression.
A sequence of real numbers is a harmonic progression (HP) if any term in the sequence is the harmonic mean of its two neighbors.
A sequence is a harmonic progression if and only if its terms are the reciprocals of an arithmetic progression that doesn't contain
The harmonic mean of and is
which proves the "if" part of the statement. The "only if" part is similar and left as an exercise: write the first two terms as and and show that the third term is the fourth term is and so on by induction.
If the term of an HP is and the term of the same HP is , what is the maximum possible length of the HP?
The reciprocals of the HP form an arithmetic progression . Then and
This gives , Note that so the arithmetic progression can only have terms before it contains So the answer is
Exercise: If the term is and the term is the answer is
If the first terms of a harmonic progression are and find the maximum partial sum
The terms of the HP are
So the maximum partial sum is
The next section will give a general approximation for partial sums of harmonic progressions.
If the sum of the first terms of an HP is , the sum of the next terms is , and the sum of the following terms is , find the sum of the following terms (again).
According to the problem statement,
Solving any two of and simultaneously gives and .
Therefore, the answer is
The partial sums of a harmonic progression are often of interest. For example, the harmonic numbers are partial sums of the harmonic progression . The following approximation for the partial sums of a harmonic progression is due to Brillianteer Aneesh Kundu.
The goal is to find the sum of the terms where .
Consider the function , and consider Riemann sums with rectangles of width starting from to .
Here, the rectangle has length , and the width of each rectangle is a constant . So the sum of the areas of these rectangles will be approximately equal to the area under the curve.
That is, the area under the curve from to is approximately
where and .
This formula gives a good approximation when is very big as compared to
The example from the previous section involved computing Apply the formula to the parenthetical sum, and rearrange terms so that is positive, i.e.
The formula gives
This is indeed a reasonable approximation to the exact value given above.
Another nice approximation of can be obtained in the following way:
Taking , note that
The first term can be obtained from the digamma function as
where is the Euler-Mascheroni constant, which is defined as
where is the harmonic number, and thus simplifies the third term. So, we get
Thus, for large , we get the approximation .