Problem 10
In the following exercises, compute the general term an of the series with the given partial sum Sn. If the sequence of partial sums converges, find its limit S. $$ S_{n}=\frac{n(n+1)}{2}, n \geq 1 $$
Problem 10
State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{\sqrt{n+3}}\)
Problem 11
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{n^{1.2}-1}{n^{2.3}+1} $$
Problem 11
Find an explicit formula for the \(n\) th term of the sequence satisfying \(a_{1}=0\) and \(a_{n}=2 a_{n-1}+1\) for \(n \geq 2\).
Problem 11
Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is given in the following problems. State if the ratio test is inconclusive. $$ a_{n}=n^{2} / 2^{n} $$
Problem 11
For \(\sum_{n=1}^{\infty} \frac{1}{n^{4}}\), calculate \(S_{5}\) and estimate the error \(R_{5}\).
Problem 11
State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sqrt{n+3}}{n}\)
Problem 12
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. \(a_{n}=\frac{n}{n+2}\)
Problem 12
Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is given in the following problems. State if the ratio test is inconclusive. $$ a_{n}=n^{10} / 2^{n} $$
Problem 12
In the following exercises, compute the general term an of the series with the given partial sum Sn. If the sequence of partial sums converges, find its limit S. $$ S_{n}=2-(n+2) / 2^{n}, n \geq 1 $$
