Hint:Bashyboy said:Homework Statement
[itex]\sum_{n=1}^{\infty} \frac{1}{2^n}[/itex]
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?
The Geometric Series Test is a method used to determine the convergence or divergence of a geometric series. It involves calculating the common ratio of the series and using it to determine if the series converges or diverges.
The Geometric Series Test is based on the comparison of the common ratio to 1, while the Integral Test involves using calculus to evaluate the convergence or divergence of a series. The Geometric Series Test is typically easier to use and requires less mathematical knowledge than the Integral Test.
The formula for the Geometric Series Test is:
If |r| < 1, then the geometric series ∑ ar^n converges to a/(1-r).
If |r| ≥ 1, then the geometric series diverges.
Yes, if the series converges, the formula for the sum of a geometric series can be used to find the exact sum. The formula is:
S = a / (1-r), where a is the first term and r is the common ratio.
The Geometric Series Test can only be used on geometric series, which have a constant ratio between consecutive terms. It also only applies to infinite series, not finite series. Additionally, the test may not always be able to determine the convergence or divergence of a series, and in some cases, other methods may need to be used.