What is the General Term for the Summation of x^(-1) from 0 to n?

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Discussion Overview

The discussion revolves around finding a general term for the summation of \( x^{-1} \) from 0 to \( n \), specifically exploring the series \( 1^{-1} + 2^{-1} + 3^{-1} + \ldots + n^{-1} \). The scope includes mathematical reasoning and exploration of related concepts such as harmonic numbers and divergence rates.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant requests a general term for the summation of \( x^{-1} \) from 0 to \( n \).
  • Another participant references harmonic numbers as a related concept.
  • A later reply suggests that there is no exact expression for the summation, noting it is approximately \( \ln(n) \).
  • One participant expresses confusion about the results and acknowledges the complexity of the topic.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a specific general term for the summation, and there are multiple viewpoints regarding the nature of the expression and its divergence.

Contextual Notes

There is mention of the rate of divergence and a formula for the \( k \)-th partial sum, but the details of these mathematical steps remain unresolved.

spectrum123
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here is the Q. i want a general term for Ʃx^(-1) for which limits are x from 0 to n
or simply a general term for 1-1+2-1+3-1+4-1+... till n
 
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ok thanks but i didn't get the result?
 
spectrum123 said:
ok thanks but i didn't get the result?

There is no exact expression - it is ≈ ln(n).
 
The section on Rate of divergence gives a formula for the kth partial sum:
4f1b3c25918c64066f4e69245ada8e56.png
 
ok now i know its out of my domain... thanks for helping
 

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