91Ó°ÊÓ

Use the estimate \(R_{N} \leq \int_{N}^{\infty} f(t) d t\) to find a bound for the remainder \(R_{N}=\sum_{n=1}^{\infty} a_{n}-\sum_{n=1}^{N} a_{n}\) where \(a_{n}=f(n) .\) $$ \sum_{n=1}^{100} n / 2^{n} $$

[T] Find the minimum value of \(N\) such that the remainder estimate \(\int_{N+1}^{\infty} f

[T] Find the minimum value of \(N\) such that the remainder estimate \(\int_{N+1}^{\infty} f

[T] Find the minimum value of \(N\) such that the remainder estimate \(\int_{N+1}^{\infty} f

[T] Find the minimum value of \(N\) such that the remainder estimate \(\int_{N+1}^{\infty} f

[T] Find the minimum value of \(N\) such that the remainder estimate \(\int_{N+1}^{\infty} f

In the following exercises, find a value of \(N\) such that \(R_{N}\) is smaller than the desired error. Compute the corresponding sum \(\sum_{n=1}^{N} a_{n}\) and compare it to the given estimate of the infinite series. \(a_{n}=\frac{1}{n^{11}}\) error \(<10^{-4}\) \(\sum_{n=1}^{\infty} \frac{1}{n^{11}}=1.000494 \ldots\)

In the following exercises, find a value of \(N\) such that \(R_{N}\) is smaller than the desired error. Compute the corresponding sum \(\sum_{n=1}^{N} a_{n}\) and compare it to the given estimate of the infinite series. \(a_{n}=\frac{1}{e^{n}}\) error \(<10^{-5}\) \(\sum_{n=1}^{\infty} \frac{1}{e^{n}}=\frac{1}{e-1}=0.581976 \ldots\)

In the following exercises, find a value of \(N\) such that \(R_{N}\) is smaller than the desired error. Compute the corresponding sum \(\sum_{n=1}^{N} a_{n}\) and compare it to the given estimate of the infinite series. \(a_{n}=\frac{1}{e^{n^{2}}}\) error \(<10^{-5}\) \(\sum_{n=1}^{\infty} n / e^{n 2}=0.40488139857 \ldots\)

In the following exercises, find a value of \(N\) such that \(R_{N}\) is smaller than the desired error. Compute the corresponding sum \(\sum_{n=1}^{N} a_{n}\) and compare it to the given estimate of the infinite series. \(a_{n}=1 / n^{4}, \quad\) error \(\quad<10^{-4}\) \(\sum_{n=1}^{\infty} 1 / n^{4}=\pi^{4} / 90=1.08232 \ldots\)