91影视

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty} \sin (n \pi / 2) \sin (1 / n)\)

Evaluate the following telescoping series or state whether the series diverges. $$ \sum_{n=1}^{\infty}(\sin n-\sin (n+1)) $$

Express the following series as a telescoping sum and evaluate its nth partial sum. $$ \sum_{n=1}^{\infty} \ln \left(\frac{n}{n+1}\right) $$

Use the root and limit comparison tests to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges. $$ \left.a_{n}=1 / x_{n}^{n} \text { where } x_{n+1}=\frac{1}{2} x_{n}+\frac{1}{x_{n}}, x_{1}=1 \text { (Hint: Find limit of }\left\\{x_{n}\right\\} .\right) $$

Express the following sums as \(p\) -series and determine whether each converges. \(\sum_{n=1}^{\infty} 2^{-\ln n}\left(\right.\) Hint: \(\left.2^{-\ln n}=1 / n^{\ln 2}\right)\)

Does \(\sum_{n=1}^{\infty} 2^{-\ln \ln n}\) converge? (Hint: Write \(2^{\ln \ln n}\) as a power of \(\left.\ln n .\right)\)

In each of the following problems, use the estimate \(\left|R_{N}\right| \leq b_{N+1}\) to find a value of \(N\) that guarantees that the sum of the first \(N\) terms of the alternating series \(\sum_{n=1}^{\infty}(-1)^{n+1} b_{n}\) differs from the infinite sum by at most the given error. Calculate the partial sum \(S_{N}\) for this \(N\).[T] \(b_{n}=1 / n\), error \(<10^{-5}\)

Determine whether the sequence defined as follows has a limit. If it does, find the limit. \(a_{1}=3, a_{n}=\sqrt{2 a_{n-1}}, n=2,3, \ldots\)

In each of the following problems, use the estimate \(\left|R_{N}\right| \leq b_{N+1}\) to find a value of \(N\) that guarantees that the sum of the first \(N\) terms of the alternating series \(\sum_{n=1}^{\infty}(-1)^{n+1} b_{n}\) differs from the infinite sum by at most the given error. Calculate the partial sum \(S_{N}\) for this \(N\).[T] \(b_{n}=1 / \ln (n), n \geq 2\), error \(<10^{-1}\)

Express the following sums as \(p\) -series and determine whether each converges. \(\sum_{n=1}^{\infty} 3^{-\ln n}\left(\right.\) Hint: \(\left.3^{-\ln n}=1 / n^{\ln 3}\right)\)