Provide an appropriate response.
-$\text { If } \sum a _ { n } \text { converges, what can be said about } \sum a _ { n } a _ { n + 1 } \text { ? }$
A) The series always diverges.
B) The series may converge or diverge.
C) The series always converges.

Question

Quizplus · Accepted Answer

The convergence or divergence of $\sum a_n a_{n+1}$ cannot be determined solely based on the convergence of $\sum a_n$. For example, consider the sequences $a_n = \frac{1}{n}$ and $b_n = (-1)^n\frac{1}{n}$. Both $\sum a_n$ and $\sum b_n$ converge, but $\sum a_na_{n+1}$ diverges ($\sum a_na_{n+1} = \sum \frac{1}{n(n+1)} = \sum\left(\frac{1}{n}-\frac{1}{n+1}\right) = 1$), while $\sum b_nb_{n+1}$ converges ($\sum b_nb_{n+1} = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$).

Provide an Appropriate Response If ∑an converges, what can be said about ∑anan+1 ? \text { If } \sum a _ { n } \text { converges, what can be said about } \sum a _ { n } a _ { n + 1 } \text { ? } If ∑an​ converges, what can be said about ∑an​an+1​ ?

Provide an Appropriate Response $\text { If } \sum a _ { n } \text { converges, what can be said about } \sum a _ { n } a _ { n + 1 } \text { ? }$