I need help with this. Please also include a detailed explanation this topic is really new to me and I don't understand it to its fullest, and I think an explanation could help. Thank you so much!<3
Let \(S\) be the set of all real numbers of the form
\(\frac{a_1}{3} + \frac{a_2}{3^2} + \frac{a_3}{3^3} + \dotsb\)
where \(a_i\) is equal to either \(1\) or \(-1\) for each \(i\)
(a) Is the number \(\frac{1}{4}\) in the set \(S\)
(b) Is the number \(\frac{1}{7}\) in the set \(S\)
Thank you so much for your help!
(a) No, 1/4 is not in the set S.
The number 1/4 can be represented as a geometric series with first term 1/4 and common ratio -1/3. However, the terms of the geometric series in the set S are all either 1/3 or -1/3. Therefore, 1/4 cannot be represented as a sum of terms in the set S.
(b) Yes, 1/7 is in the set S.
The number 1/7 can be represented as a geometric series with first term 1/7 and common ratio -1/3. The terms of the geometric series in the set S are all either 1/3 or -1/3. Therefore, 1/7 can be represented as a sum of terms in the set S.
