Why would you want to use the root test? It looks like you're using the ratio test. As Dustinfls suggests, this one looks ripe for L'Hopital's Rule.mattmannmf said:Using the root test find whether the series converges or diverges:
Lim Sin (4/(3n+3)) / Sin (4/(3n))
n-> inf.
I have no idea how to cancel out the sin terms
The root test is a method used to determine if a series converges or diverges by examining the limit of the nth root of the absolute value of the terms in the series.
To use the root test, you take the nth root of the absolute value of each term in the series and then take the limit as n approaches infinity. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.
The root test is most useful for series with positive terms, including series with alternating signs.
No, the root test can only be used for series with non-negative terms. It cannot be used for series with negative terms or series with alternating signs.
Both the root test and the ratio test are used to determine the convergence or divergence of a series. The main difference is that the root test uses the nth root of the terms while the ratio test uses the ratio of consecutive terms. The root test is often easier to use for series with alternating signs, while the ratio test is more useful for series with factorial terms.