Question

Show that n=1 to ∞ ∑(-1)^n+1 5/7n+2 is conditionally convergent.

Answer 1

$\sum_{n=1}^{\infty}{\left( -1 \right) ^{n+1}\frac{5}{7n+2}}\,\,-\,\,Leibnitz\,\,series:\\altering\,\,signs;\\\frac{5}{7n+2}\downarrow 0,n\rightarrow \infty \\Hence\,\,the\,\,series\,\,converges.\\The\,\,series\ \sum_{n=1}^{\infty}{\frac{5}{7n+2}}\\does\,\,not\,\,converge\,\,by\,\,the\,\,comparison\,\,test:\\\underset{n\rightarrow \infty}{\lim}\frac{\frac{5}{7n+2}}{\frac{1}{n}}=\underset{n\rightarrow \infty}{\lim}\frac{5}{7+\frac{2}{n}}=\frac{5}{7}\\and\,\,\sum_{n=1}^{\infty}{\frac{1}{n}}\,\,does\,\,not\,\,converge\\Hence\,\,the\,\,series\,\,is\,\,conditionally\,\,convergent$