Questions: Find the sum of the convergent series. 25-5+1-(1/5)+...

Solution Steps

To find the sum of the convergent series, we first identify the type of series. The given series appears to be a geometric series with the first term \( a = 25 \) and a common ratio \( r = -\frac{1}{5} \). A geometric series converges if the absolute value of the common ratio is less than 1. The sum of an infinite convergent geometric series can be calculated using the formula \( S = \frac{a}{1 - r} \).

The given series is \( 25 - 5 + 1 - \frac{1}{5} + \ldots \). This is a geometric series where the first term \( a = 25 \) and the common ratio \( r = -\frac{1}{5} \).

A geometric series converges if \( |r| < 1 \). Here, \( |r| = \frac{1}{5} < 1 \), so the series converges.

The sum \( S \) of an infinite convergent geometric series is given by the formula: \[ S = \frac{a}{1 - r} \] Substituting the values: \[ S = \frac{25}{1 - \left(-\frac{1}{5}\right)} = \frac{25}{1 + \frac{1}{5}} = \frac{25}{\frac{6}{5}} = 25 \cdot \frac{5}{6} = \frac{125}{6} \approx 20.8333 \]

Final Answer