- #1
phosgene
- 146
- 1
Homework Statement
Use the limit comparison test to determine whether the following series converges or diverges.
[itex]\sum{\frac{2^n}{3^n - 1}}[/itex]
Where the sum is from n = 1 to n = ∞.
Homework Equations
The limit comparison test:
Suppose an>0 and bn>0 for all n. If the limit of an/bn=c, where c>0, then the two series [itex]\sum{a_{n}}[/itex] and [itex]\sum{b_{n}}[/itex] either both converge or both diverge.
The Attempt at a Solution
So I get that the point is to find another series that you know diverges or converges that will also make the above limit easy to evaluate. All I can think to do is to compare it to something like [itex]\sum{1/3^n}[/itex] or [itex]\sum{2^n}[/itex], the first of which results in the limit not existing and the second of which results in the limit being zero. I think it's obvious that the series should converge, as the denominator is always equal to or greater than the numerator. But I'm out of ideas on what other sums I could use in the limit comparison test.