MHB Which is Smaller: Comparing Mathematical Expressions A and B?

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The mathematical expressions A and B are compared, with A defined as an alternating series and B as a sum of positive fractions. It is shown that A can be expressed as the difference between harmonic numbers, specifically A = H_2014 - H_1007. This leads to the conclusion that A equals the sum of fractions from 1/1008 to 1/2014. Ultimately, the analysis concludes that A is smaller than B.
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$A=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+----+\dfrac{1}{2013}-\dfrac{1}{2014}$

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

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