Answer to Question #109845 in Real Analysis for Pappu Kumar Gupta

Question

Test the absolute and conditional convergence of the series summation n=1 to infinity (-1)^n/2n+7

Answer 1

Let's, first of all, prove that there is no absolute convergence: the series of absolute values is convergent. But the series, where a_n = 1/(2n+7) is divergent because 1/(2n+7) is asymptotically equivalent to 1/(2n) and the series with terms c_n=1/(2n) is divergent. Here we used the fact that if a_n is asymptotically equivalent to the C/n^p, where C is some constant and p is less or equal to the 1, then the respective series is divergent.

Let's now show that there is conditional convergence. We shall use the Dirichlet's test.

The sum of (-1)ⁿ where n=1,2,..,M (where M is some constant) is always less or equal to the 1, ((-1 + (-1)² + (-1)³ +..+ (-1)^M)≤1. The sequence with a_n=1/(2n+7) is monotonic and tends to the 0, when n tends to the infinity. By the Dirichlet's test: the series with the common member a_nb_n is convergent. Exactly what we needed.