Given:
[tex]\frac{1}{4}+\frac{1}{3}+\frac{4}{9}+\frac{16}{27}+\frac{64}{81}+.......[/tex]
Required:
To find whether the given series is convergent or divergent.
Explanation:
The given series is geometric series.
And the common ratio r is,
[tex]\begin{gathered} r=\frac{\frac{1}{3}}{\frac{1}{4}} \\ \\ =\frac{4}{3} \end{gathered}[/tex]
As
[tex]|r|\ge1[/tex]
that means the terms do not tend towards 0, as
[tex]n\rightarrow\infty[/tex]
Therefore, the series is divergent.
Final Answer:
The given series is divergent, because the series is geometric and the absolute value of the common ratio is greater than 1.
