Seriously? You are aware, are you not, that (-3)^(n+1)/(-3)^n= -3? The (4^n part is just as easy! You should not need L'Hospital's rule.superdave said:Homework Statement
Does this series converge or diverge?
Series from n=1 to infinity n(-3)^(n+1) / 4^(n-1)
Homework Equations
The Attempt at a Solution
Okay, I've tried it both ways.
Ratio test:
lim n --> inf. ((n+1)*(-3)^(n+1)/4^n) / (n * (-3)^n / 4^(n-1))
Now, that doesn't appear to simplify in anyway that would make using l'hospital's rule possible to find the limit.
Root test:
lim n --> inf. of -3*n^(1/n) / 4^((n-1)/n)
That bottom part throws me off.
The root test for divergence is a method used to determine the convergence or divergence of a series. It involves taking the nth root of the absolute value of each term in the series and then taking 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 and another method should be used.
The root test is used by taking the nth root of each term in the series and then taking the limit as n approaches infinity. This limit is then compared to 1. If the limit is less than 1, the series converges and if the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive and another method should be used.
The ratio test for divergence is another method used to determine the convergence or divergence of a series. It involves taking the absolute value of the ratio of each term in the series to the next term and then taking 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 and another method should be used.
The ratio test is used by taking the absolute value of the ratio of each term in the series to the next term and then taking the limit as n approaches infinity. This limit is then compared to 1. If the limit is less than 1, the series converges and if the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive and another method should be used.
The root test should be used when the series contains nth roots or powers, while the ratio test should be used when the series contains factorial or exponential terms. In general, the ratio test is more versatile and can be used in a wider range of situations. Both tests should be used when the series contains both nth roots and factorial terms.