Questions: What is the sum of the series from n=1 to infinity of 2(1/5)^(n-1)? S=1/5 S=2/5 S=5/3 S=5/2

Solution Steps

To find the sum of the infinite series $\sum_{n=1}^{\infty} 2\left(\frac{1}{5}\right)^{n-1}$, we recognize that this is a geometric series with the first term $a = 2$ and common ratio $r = \frac{1}{5}$. The sum of an infinite geometric series is given by the formula $S = \frac{a}{1 - r}$.

The given series is $\sum_{n=1}^{\infty} 2\left(\frac{1}{5}\right)^{n-1}$. This is a geometric series with:

• First term $a = 2$
• Common ratio $r = \frac{1}{5}$

The sum $S$ of an infinite geometric series is given by: $$S = \frac{a}{1 - r}$$

Substitute $a = 2$ and $r = \frac{1}{5}$ into the formula: $$S = \frac{2}{1 - \frac{1}{5}} = \frac{2}{\frac{4}{5}} = 2 \times \frac{5}{4} = 2.5$$

Final Answer