Question

Find whether the following series are convergent or divergent:
$\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...$

Answer 1

To find whether the following series is convergent or divergent

Given series is

$\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+....$

This can be rewritten as

$(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+......$

Here,

$a_1=1$

$a_2=\frac{1}{2}$

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As 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 the terms decrease indefinitely.

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

Therefore, from the above statement given series converges.