Question

Find whether the following series is convergent or divergent:
$\frac{1}{xy}-\frac{1}{(x+1)(y+1)}+\frac{1}{(x+2)(y+2)}-\frac{1}{(x+3)(y+3)}+....,$ $x$ and $y$ being positive quantities.

Answer 1

To find whether the following series is convergent or divergent

Given series is

$\frac{1}{xy}-\frac{1}{(x+1)(y+1)}+\frac{1}{(x+2)(y+2)}-\frac{1}{(x+3)(y+3)}+....$

where $x$ and $y$ being positive quantities

This can also be written as

$=\frac{1}{y-x}[(\frac{1}{x}-\frac{1}{y})+(\frac{1}{x+1}-\frac{1}{y+1})+(\frac{1}{x+2}-\frac{1}{y+2})+........]$

Here,

$a_1=\frac{1}{xy}$

$a_2=\frac{1}{(x+1)(y+1)}$

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We know that, an infinite series in which we have the terms are alternatively positive and negative is convergent if each term is numerically less than the preceding and if the terms decrease indefinitely.

Since,

$a_1>a_2>a_3>a_4>a_5.....$

Therefore, from the above statement given series converges.