Question

Determine whether the series is absolutely convergent, conditionally convergent,or divergent.

\(\sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{5n + 1}}} \)

Short answer

The Series is Conditionally Convergent..

Step-by-step solution

(Using Leibnitz Test)

\({U_n} > {U_{n + 1}}\)

And\({\lim _{n \to \infty }}{u_n} = \frac{{{{( - 1)}^n}}}{{5n + 1}} = 0\)

So, The Series is Convergent

(Test for absolute or conditionally convergent)

\(|{u_n}| = \frac{1}{{5n + 1}}\)

By the comparison test

\({v_n} = \frac{1}{n}\)

\(\frac{{{u_n}}}{{{v_n}}} = \frac{n}{{5n + 1}}\)

\({\lim _{n \to \infty }}\frac{{{u_n}}}{{{v_n}}} = \frac{1}{5}\)=Finite number

But by the P Series Test\({v_n}\)is divergent.

Hence, the series is Divergent..

Hence, The Given series is Conditionally Convergent..