Question

show that∑n=0∞ (-1)^(n+1)(5)/(7n+2)` is conditionally convergent 

Answer 1

Consider the series

$$\displaystyle\sum_{n=0}^{\infin}|\dfrac{(-1)^{n+1}}{7n+2}|=\displaystyle\sum_{n=0}^{\infin}\dfrac{1}{7n+2}$$

Use the Integral Test

Let $f(n)=a_n=\dfrac{1}{7n+2}, n=0,1,2,...$

$$\displaystyle\int_{0}^{\infin}\dfrac{1}{7x+2}dx=\lim\limits_{t\to\infin}\displaystyle\int_{0}^{t}\dfrac{1}{7x+2}dx$$

$$=\lim\limits_{t\to\infin}\dfrac{1}{7}[\ln|7x+2|]\begin{matrix} t \\ 0 \end{matrix}=\infin$$

Since this improper integral is divergent, the series $\displaystyle\sum_{n=0}^{\infin}\dfrac{1}{7n+2}$ diverges by the Integral Test.

Consider the alternating series

$$\displaystyle\sum_{n=0}^{\infin}\dfrac{(-1)^{n+1}}{7n+2}$$ $$b_n=\dfrac{1}{7n+2}>0, n=0,1,2,...$$

$$b_{n+1}=\dfrac{1}{7(n+1)+2}<\dfrac{1}{7n+2}=b_n, n=0, 1, 2, ...$$

$$\lim\limits_{n\to \infin}b_n=\lim\limits_{n\to \infin}\dfrac{1}{7n+2}=0$$

Then the series $\displaystyle\sum_{n=0}^{\infin}\dfrac{(-1)^{n+1}}{7n+2}$ is convergent by the Alternating Series Test.

Therefore the series $\displaystyle\sum_{n=0}^{\infin}\dfrac{(-1)^{n+1}}{7n+2}$ is conditionally convergent.