Question

The Root and Ratio Test. We want to determine whether the series $\sum_{n=1}^{\infty} (-1)^n \left( 1 - \frac{1}{n} \right)^{n^2}$ converges or diverges. Since $a_n = (-1)^n \left( 1 - \frac{1}{n} \right)^{n^2}$ contains a exponent of $n$, we have decided to apply the Root Test and find that, $\rho = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty}$ Note: type a simplified ratio in terms of $n$, then evaluate the limit above. Since $\rho$ choose one $1$ we conclude that choose one Note: You can earn partial credit on this problem.

          The Root and Ratio Test.
We want to determine whether the series $\sum_{n=1}^{\infty} (-1)^n \left( 1 - \frac{1}{n} \right)^{n^2}$ converges or diverges.
Since $a_n = (-1)^n \left( 1 - \frac{1}{n} \right)^{n^2}$ contains a exponent of $n$, we have decided to apply the Root Test and find that,
$\rho = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty}$ 
Note: type a simplified ratio in terms of $n$, then evaluate the limit above.
Since $\rho$ choose one $1$ we conclude that choose one
Note: You can earn partial credit on this problem.

The Root and Ratio Test.
We want to determine whether the series ∑n=1^∞ (-1)^n ( 1 - (1)/(n))^n^2 converges or diverges.
Since an = (-1)^n ( 1 - (1)/(n))^n^2 contains a exponent of n, we have decided to apply the Root Test and find that,
ρ = limn →∞√(|an|) = limn →∞
Note: type a simplified ratio in terms of n, then evaluate the limit above.
Since ρ choose one 1 we conclude that choose one
Note: You can earn partial credit on this problem.

Added by Leslie C.

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