whatlifeforme said:Homework Statement
Determine if series converges or diverges.
Homework Equations
[itex]\displaystyle \sum^{∞}_{n=0} \frac{cos n* \pi}{5^n}[/itex]The Attempt at a Solution
I have tried using the nth term test but not sure how to take the limit of cosine.
HallsofIvy said:First, [itex]cos(n\pi)[/itex] is just [itex](-1)^n[/itex] so this is [itex]\frac{(-1)^n}{5^n}[/itex] and the ratio test works nicely.
The definition of convergence for a series is when the partial sums of the terms approach a finite limit as the number of terms increases. This means that the terms in the series get smaller and closer to a particular value as more terms are added.
To determine if a series converges or diverges, you can use several tests such as the comparison test, the ratio test, and the integral test. These tests involve analyzing the behavior of the series as the number of terms increases.
A series is absolutely convergent if the absolute values of its terms converge. On the other hand, a series is conditionally convergent if it converges but not absolutely. This means that the series can be rearranged to have a different sum.
The limit comparison test is used to compare the behavior of a given series with that of a known, simpler series. If the limit of the ratio between the two series is a positive finite value, then both series have the same behavior and will either both converge or both diverge.
No, a series must have a single, definite sum for it to be considered convergent. If a series diverges, it means that the sum does not approach a finite value and therefore cannot converge at any point. However, it is possible for a series to have a conditional convergence, which allows for different rearrangements of the terms to converge to different values.