Paper 4, Section II, E

What does it mean to say that the sequence of real numbers $\left(x_{n}\right)$ converges to the limit $x ?$ What does it mean to say that the series $\sum_{n=1}^{\infty} x_{n}$ converges to $s$ ?

Let $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ be convergent series of positive real numbers. Suppose that $\left(x_{n}\right)$ is a sequence of positive real numbers such that for every $n \geqslant 1$, either $x_{n} \leqslant a_{n}$ or $x_{n} \leqslant b_{n}$. Show that $\sum_{n=1}^{\infty} x_{n}$ is convergent.

Show that $\sum_{n=1}^{\infty} 1 / n^{2}$ is convergent, and that $\sum_{n=1}^{\infty} 1 / n^{\alpha}$ is divergent if $\alpha \leqslant 1$.

Let $\left(x_{n}\right)$ be a sequence of positive real numbers such that $\sum_{n=1}^{\infty} n^{2} x_{n}^{2}$ is convergent. Show that $\sum_{n=1}^{\infty} x_{n}$ is convergent. Determine (with proof or counterexample) whether or not the converse statement holds.

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