91影视

For each of the following sequences, if the divergence test applies, either state that \(\lim _{n \rightarrow \infty} a_{n}\) does not exist or find \(\lim _{n \rightarrow \infty} a_{n} .\) If the divergence test does not apply, state why. \(a_{n}=\frac{1-\cos ^{2}(1 / n)}{\sin ^{2}(2 / n)}\)

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \ln \left(1+\frac{1}{n}\right)\)

Plot the first \(N\) terms of each sequence. State whether the graphical evidence suggests that the sequence converges or diverges. $$ \text { [7] } a_{1}=1, a_{2}=2, a_{3}=3, \text { and for } n \geq 4, a_{n}=\sqrt{a_{n-1} a_{n-2} a_{n-3}} ; N=30 $$

State whether the given series converges and explain why. $$ \sum_{n=1}^{\infty} \frac{1}{n+10^{80}} \text { (Hint: Rewrite using a change of index.) } $$

Use the limit comparison test to determine whether each of the following series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1+1 / n}} $$

Use the root test to determine whether \(\sum_{m=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$ a_{n}=\frac{(\ln n)^{2 n}}{n^{n}} $$

Suppose that \(\lim _{n \rightarrow \infty} a_{n}=1, \lim _{n \rightarrow \infty} b_{n}=-1\), and \(0<-b_{n}

Use the root test to determine whether \(\sum_{m=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$ a_{n}=n / 2^{n} $$

Use the limit comparison test to determine whether each of the following series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{1}{2^{1+1 / n} n^{1+1 / n}} $$

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{1+n^{4}}\)