Question

Find whether the following series are convergent or divergent:
$\frac{1}{x}-\frac{1}{x+a}+\frac{1}{x+2a}-\frac{1}{x+3a}+....,x$ and $a$ being positive quantities.

Answer 1

To find Whether the following series are Convergent or divergent

$\frac{1}{x}-\frac{1}{x+a}+\frac{1}{x+2a}-\frac{1}{x+3a}+.....,$

Where $x$ and $a$ being positive quantities

An infinite series in which the terms are alternatively positive and negative is convergent if each term is numerically less than the preceding term and if the terms decrease indefinitely

Here,

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

$a_2=\frac{1}{x+a}$

$a_3=\frac{1}{x+2a}$

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

Therefore, from the above statement given series converges.