Which is Smaller: Comparing Mathematical Expressions A and B?

[Albert1]

$A=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+----+\dfrac{1}{2013}-\dfrac{1}{2014}$

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

 

[chisigma]

$A=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+----+\dfrac{1}{2013}-\dfrac{1}{2014}$

$B=\dfrac{1}{1007}+\dfrac{1}{1008}+----+\dfrac{1}{2013}+\dfrac{1}{2014}$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

$\chi$ $\sigma$

 

[kaliprasad]

[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

$\chi$ $\sigma$

Spoiler

$\displaystyle A = H_{2014} - H_{1007}$

 

[Albert1]

$A=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+----+\dfrac{1}{2013}-\dfrac{1}{2014}$

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

Spoiler

let $C=1+\dfrac{1}{2}+\dfrac{1}{3}+----+\dfrac{1}{1007}$
then :
$A+C=1+\dfrac{1}{2}+\dfrac{1}{3}+---+\dfrac{1}{1007}+\dfrac{1}{1008}+---+\dfrac{1}{2013}+\dfrac{1}{2014}$
$B+C=1+\dfrac{1}{2}+\dfrac{1}{3}+---+\dfrac{1}{1007}+\dfrac{1}{1007}+\dfrac{1}{1008}+---+\dfrac{1}{2013}+\dfrac{1}{2014}$
$B+C-(A+C)=B-A=\dfrac{1}{1007}$
$\therefore B>A$