Which is Smaller: Comparing Mathematical Expressions A and B?

  • Context: MHB 
  • Thread starter Thread starter Albert1
  • Start date Start date
Click For Summary

Discussion Overview

The discussion revolves around comparing two mathematical expressions, A and B, defined by alternating series and sums of reciprocals. Participants are exploring the values of these expressions and attempting to determine which is smaller. The scope includes mathematical reasoning and analysis of series.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants express that A can be rewritten in terms of harmonic numbers, suggesting that A equals the difference between two harmonic sums: \( H_{2014} - H_{1007} \).
  • Others propose that A can also be expressed as \( H_{2013} - H_{1007} \), leading to a similar conclusion about the relationship between A and B.
  • There are repeated assertions that A is less than B based on these formulations, although the reasoning and steps are not universally agreed upon.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the comparison between A and B, as different formulations and interpretations of A lead to conflicting conclusions about which expression is smaller.

Contextual Notes

The discussion includes potential ambiguities in the definitions of harmonic numbers and the limits of the series involved, which may affect the comparison. Some steps in the reasoning are not fully resolved.

Albert1
Messages
1,221
Reaction score
0
$A=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+----+\dfrac{1}{2013}-\dfrac{1}{2014}$

$B=\dfrac{1}{1007}+\dfrac{1}{1008}+----+\dfrac{1}{2013}+\dfrac{1}{2014}$Compare A and B,which on is smaller ?
 
Mathematics news on Phys.org
Albert said:
$A=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+----+\dfrac{1}{2013}-\dfrac{1}{2014}$

$B=\dfrac{1}{1007}+\dfrac{1}{1008}+----+\dfrac{1}{2013}+\dfrac{1}{2014}$Compare A and B,which on is smaller ?

[sp]Is...

$\displaystyle A = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{2013} - \frac{1}{2014} = H_{2014} - H_{1007} =$

$ \displaystyle = \frac{1}{1008} + \frac{1}{1009} + ... + \frac{1}{2014}\ (1)$

... so that is A < B...[/sp]

Kind regards

$\chi$ $\sigma$
 
Last edited:
chisigma said:
[sp]Is...

$\displaystyle A = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{2013} - \frac{1}{2014} = H_{2013} - H_{1007} =$

$ \displaystyle = \frac{1}{1008} + \frac{1}{1009} + ... + \frac{1}{2013}\ (1)$

... so that is A < B...[/sp]

Kind regards

$\chi$ $\sigma$
$\displaystyle A = H_{2014} - H_{1007}$
 
Albert said:
$A=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+----+\dfrac{1}{2013}-\dfrac{1}{2014}$

$B=\dfrac{1}{1007}+\dfrac{1}{1008}+----+\dfrac{1}{2013}+\dfrac{1}{2014}$Compare A and B,which on is smaller ?
let $C=1+\dfrac{1}{2}+\dfrac{1}{3}+----+\dfrac{1}{1007}$
then :
$A+C=1+\dfrac{1}{2}+\dfrac{1}{3}+---+\dfrac{1}{1007}+\dfrac{1}{1008}+---+\dfrac{1}{2013}+\dfrac{1}{2014}$
$B+C=1+\dfrac{1}{2}+\dfrac{1}{3}+---+\dfrac{1}{1007}+\dfrac{1}{1007}+\dfrac{1}{1008}+---+\dfrac{1}{2013}+\dfrac{1}{2014}$
$B+C-(A+C)=B-A=\dfrac{1}{1007}$
$\therefore B>A$
 

Similar threads

MHB Find Product: $(a^4+b^4+c^4)$ and $(a^6+b^6+c^6)$
  • · Replies 3 ·
Replies
3
Views
1K
MHB Proving $(a)^\dfrac{1}{3}+(b)^\dfrac{1}{3}+(c)^\dfrac{1}{3}=0$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Roots of Polynomial: Find $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Prove $\frac{1}{3}\le \frac{u}{v} \le \frac{1}{2}$ with Triangle Sides
  • · Replies 4 ·
Replies
4
Views
1K
MHB Proving $abcd\ge 3$ with $a, b, c, d>0$
  • · Replies 1 ·
Replies
1
Views
1K
MHB What is the remainder when m+n is divided by 1000 in a trigonometric challenge?
  • · Replies 2 ·
Replies
2
Views
1K
MHB Finding Min Value of $\dfrac{|b|+|c|}{a}$ from Roots of Cubic Equations
  • · Replies 4 ·
Replies
4
Views
2K
MHB Can You Solve This Challenging Inequality Problem?
  • · Replies 1 ·
Replies
1
Views
2K
MHB How Can You Rewrite an Equation to Express y as a Function of x?
  • · Replies 3 ·
Replies
3
Views
1K
MHB Comparing Trigonometric and Fractional Values: Which is Greater?
  • · Replies 4 ·
Replies
4
Views
2K