1. ## real analysis

1. If ∑an with an >0 is convergent, then is ∑an² always convergent? Either prove it or give a counterexample?
2. If ∑an with an >0 is convergent, then is ∑(anan+1)^(1/2) always convergent? Either prove it or give a counterexample?
3. Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent.
Thanks

2. Originally Posted by yuan.69
1. If ∑an with an >0 is convergent, then is ∑an² always convergent? Either prove it or give a counterexample?
If $\sum a_n$ converges, it implies that $\lim_{n \to \infty} a_n=0$
Hence, $\forall \epsilon >0 ~,~ \exists N \in \mathbb{N} ~,~ \forall n \geqslant N, a_n<\epsilon$
now consider $\epsilon=1$ and consider that we are working on $n \geqslant N$
then $a_n^2 \leqslant a_n$

thus $\sum_{n \geqslant N} a_n^2 \leqslant \sum_{n \geqslant N} a_n$

does this help ?

3. Originally Posted by yuan.69
2. If ∑an with an >0 is convergent, then is ∑(anan+1)^(1/2) always convergent? Either prove it or give a counterexample?
By AM–GM, $\left(a_na_{n+1}\right)^{\frac12}\leqslant\frac{a_ n+a_{n+1}}2$

$\therefore\ \sum_{n\,=\,1}^\infty\left(a_na_{n+1}\right)^{\fra c12}\ \leqslant\ \sum_{n\,=\,1}^\infty\left(\frac{a_n+a_{n+1}}2\rig ht)=\frac{a_1}2+\sum_{n\,=\,2}^\infty a_n$

Does this help?