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I understand what Arithmetic, Geometric sequence and series are, and how to find their $n^{\, \text{th}}$ terms, sum, convergence and divergence. But I run into confusion when I try understanding "harmonic sequence and series."

This is what I have understood till now:

Harmonic sequence is a sequence where the sequence is formed by taking the reciprocal of each term of an arithmetic sequence, few examples:

  1. AP sequence: $1,2,3,4,5, \dots$ ~ it's equivalent HP sequence: $\frac{1}{1},\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}, \dots$
  2. AP sequence: $\frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{4}{3}, \frac{5}{3}, \dots$ ~ it's equivalent HP sequence: $\frac{3}{1}, \frac{3}{2}, \frac{3}{3}, \frac{3}{4}, \dots$

Similarly, if we are given a HP sequence then we can reciprocal it and get an AP sequence. So, basically another def of Harmonic sequence is that "if after taking the reciprocal of HP terms, they are in AP sequence, then the given sequence is HP."

Which means, if we are given a HP sequence and we need to find out the series, i.e sum of the terms of the HP sequence. We should be able to reciprocal the terms of HP to AP and use formulas of AP to calculate the sum or find the terms. But that isn't possible!

Also, sometimes, when talking about harmonic series ~ people just focus on "sum of all positive unit fractions." I understood the other two very well, but cannot understand HP much.

Some people say there's no formula for finding the sum of HP series, but then other's say that you just reciprocal the series and find the sum in AP. Some say we can approximate using formula or constant

I don't understand calculus! But if you have resources that talk about HP please feel free to drop some!

Pavan C.
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Pratik Hadawale
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  • The problem with the reciprocal idea is that in general $1/x+1/y\ne 1/(x+y)$ – eyeballfrog Mar 03 '23 at 13:29
  • @eyeballfrog I do understand that! like if you were to add $1/x+1/y$ it's result won't be same as adding $1/(x+y)$ or doing reciprocal of terms that is adding $x+y$ together and then taking reciprocal of the result to get sum in HP. But people do this all over youtube! – Pratik Hadawale Mar 03 '23 at 13:37
  • Yes, people are like that. What do you want us to do about it? – Ivan Neretin Mar 03 '23 at 13:39
  • @IvanNeretin I don't want you to do anything about it. But I am confused as to what Harmonic sequence and series actually is, every other youtube video explains something that is contradictory to other video or source, i want you guys to suggest me a good resource that explains harmonic sequence and series – Pratik Hadawale Mar 03 '23 at 13:42
  • For any good resource, there is another good resource that contradicts it. – Ivan Neretin Mar 03 '23 at 13:44
  • When you subtract two harmonic series at a different rate you get a beautiful result https://math.stackexchange.com/questions/46378/do-these-series-converge-to-logarithms – Jaume Oliver Lafont Mar 03 '23 at 14:08
  • @PratikHadawale Could you give an example of such a video? – eyeballfrog Mar 03 '23 at 14:19

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