Respuesta :
Answer with explanation:
The given series is
[tex]\frac{1}{5}+\frac{1}{15}+\frac{1}{45}+\frac{1}{81}+.....[/tex]
The given series is a geometric sequence,whose common ratio is equal to
[tex]R=\frac{2^{nd}\text{term}}{1^st \text{term}}\\\\R=\frac{\frac{1}{15}}{\frac{1}{5}}\\\\R=\frac{5}{15}\\\\R=\frac{1}{3}[/tex]
Sum to Infinity is given by the formula
[tex]S_{\infty}=\frac{\text{First term}}{1-\text{Common ratio}} \text{or}\frac{\text{First term}}{\text{Common ratio}-1} \\\\S_{\infty}=\frac{\frac{1}{5}}{1-\frac{1}{3}}\\\\S_{\infty}=\frac{\frac{1}{5}}{\frac{2}{3}}\\\\S_{\infty}=\frac{3}{10}[/tex]
As the sum of series is Finite, that is having a single value, so the series is Convergent.
If it has more than one sum, it would have been Divergent.
Option D: It converges; it has a sum.
