Question

Find whether the following series are convergent or divergent:
$\sqrt{\frac{1}{2}}+\sqrt{\frac{2}{3}}+\sqrt{\frac{3}{4}}+\sqrt{\frac{4}{5}}+......$

Answer 1

To find whether the following series are convergent or divergent

$\sqrt{\frac{1}{2}}+\sqrt{\frac{2}{3}}+\sqrt{\frac{3}{4}}+\sqrt{\frac{4}{5}}$

Here, $a_n=\sqrt{\frac{n}{n+1}}$

Here, we are using series divergence test

If $\underset{n\to \infty }{lim}{a}_n\ne 0$ then $\sum a_n$ Diverges

On using divergence test,

$\underset{n\to \infty }{lim}\left(\sqrt{\frac{n}{n+1}}\right)=1\ne 0$

Thus, by the divergence test criteria given series diverges.