The discussion revolves around the convergence of the series \(\sum_{n=1}^{\infty} \frac{1}{2^n}\) and the application of the geometric series test. Participants are exploring whether the series can be manipulated to fit the criteria for this test.
Participants are actively engaging with the problem, raising questions about the validity of certain assumptions regarding the series' starting index. Some guidance has been offered regarding how to adjust the series to fit the geometric series framework, but there is no explicit consensus on the correct approach.
There is mention of a potential misunderstanding regarding the starting index of the series, with references to teacher guidance that may need clarification. The implications of starting the series at \(n=1\) versus \(n=0\) are under consideration.
Bashyboy said:Homework Statement
\sum_{n=1}^{\infty} \frac{1}{2^n}
Homework Equations
The Attempt at a Solution
Could I some how manipulate this to fit a geometric series, so that I may instead use the geometric series test?
Do you mean, n can't start at 1, it must start at zero?Bashyboy said:Well, I was told by my teacher that n couldn't equal one for the geometric series to be implemented, was he, perhaps, wrong?