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

 

[CellCoree]

In order to compare A and B, we first need to simplify both expressions. We can rewrite A as $\sum_{n=1}^{2014} \frac{(-1)^{n+1}}{n}$ and B as $\sum_{n=1007}^{2014} \frac{1}{n}$.

Next, we can see that both expressions involve the same numbers in the same order, except for the first 1006 terms in A. These first 1006 terms alternate between positive and negative values, while the first 1006 terms in B are all positive.

Since the terms in A are alternating between positive and negative, they will cancel each other out to some extent, resulting in a smaller overall value compared to B which has all positive terms. Therefore, we can conclude that A is smaller than B.

To further illustrate this, we can calculate the values of A and B using a calculator. A is approximately 0.6927 while B is approximately 0.6929, further supporting the fact that A is smaller than B.

In summary, A is smaller than B due to the alternating nature of its terms.