real analysis

**yuan.69** · March 1st 2009, 07:43 AM

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

**Moo** · March 1st 2009, 07:50 AM

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 ?

**JaneBennet** · March 1st 2009, 03:11 PM

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?

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