In a geometric series, the sum of the first \( n \) terms is given by \( S_{n}=p\left(1-\left(\frac{1}{2}\right)^{n}\right) \) and the sum to infinity of this series is 10 . 5.9.1 Calculate the value of \( p \).

In a geometric series, the sum to infinity \( S \) can be expressed as \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Here, since the sum to infinity is given as 10, we can set it equal to \( \frac{p}{1 - \frac{1}{2}} = 10 \). So, \( \frac{p}{\frac{1}{2}} = 10 \) leads us to \( 2p = 10 \). Thus, we find that \( p = 5 \). Now you have found that \( p = 5 \). This means the first term of your geometric series is 5, making your series even more interesting with those terms starting to unfold! Each next term will be half of the previous one, leading towards that infinite sum of 10. Keep up the good work with your studies!