consider the following series.\n∑(n = 1 to ∞) ((-1)^(n + 1))/(n^7) (|error| < 0.00005)\nshow that the series…

Explanation:

Step1: Recall alternating - series test

The alternating - series test states that for an alternating series $\sum_{n = 1}^{\infty}(-1)^{n + 1}a_n$ (where $a_n>0$), the series converges if two conditions are met: 1. $\lim_{n\rightarrow\infty}a_n = 0$ and 2. $a_{n+1}<a_n$ for all $n$ greater than some positive integer $N$. In our case, $a_n=\frac{1}{n^7}$.

Step2: Check the limit condition

We find $\lim_{n\rightarrow\infty}a_n=\lim_{n\rightarrow\infty}\frac{1}{n^7}$. As $n$ approaches infinity, the denominator $n^7$ approaches infinity, so $\lim_{n\rightarrow\infty}\frac{1}{n^7}=0$.

Step3: Check the decreasing - sequence condition

We compare $a_n=\frac{1}{n^7}$ and $a_{n + 1}=\frac{1}{(n + 1)^7}$. Since $(n+1)^7>n^7$ for all positive integers $n$, then $\frac{1}{(n + 1)^7}<\frac{1}{n^7}$.

Answer:

$\lim_{n\rightarrow\infty}\frac{1}{(n + 1)^7}=0$, $\frac{1}{(n + 1)^7}<\frac{1}{n^7}$, $\lim_{n\rightarrow\infty}\frac{1}{n^7}=0$