n=1 series to infinity (-5^n)/n^3 does it absolutely converge, diverge or conditionally converge. Would I be applying the ratio test?
Sorry it's *(-5)^n and I don't think its absolutely convergent, am I correct?
http://www.math.hmc.edu/calculus/tutorials/convergence/
Scroll down to "Alternating series test"
thanks!
To determine whether the series converges absolutely, diverges, or conditionally converges, you are correct in considering the Ratio Test.
The Ratio Test states that if the absolute value of the ratio of consecutive terms approaches a number less than 1 as n approaches infinity, then the series converges absolutely. If the ratio approaches a number greater than 1 or if the limit is undefined, the series diverges. If the ratio approaches exactly 1 or if the ratio test is inconclusive, additional tests are usually required.
Let's apply the Ratio Test to the given series:
Consider the ratio of consecutive terms, |(-5^(n+1))/(n+1)^3| / |(-5^n)/n^3|. Simplifying this ratio, we get:
|(-5)^(n+1) * n^3| / |(-5)^n * (n+1)^3|
Rearranging the terms, we have:
|(n^3 * (-5)^(n+1)) / ((n+1)^3 * (-5)^n)|
Simplifying further, we get:
|(-5) * (n^3/n+1)^3|
As n approaches infinity, we can observe that the term (n^3/n+1)^3 approaches 1, since the exponent does not significantly impact the value. Thus, we have:
|-5|
Since |-5| < 1, the ratio of consecutive terms approaches a number less than 1. Therefore, according to the Ratio Test, the series converges absolutely.
Hence, the series (-5^n)/n^3 converges absolutely.