Lp space and sequence For what value of $p$ the sequence $\displaystyle x_{n}=\frac{1}{n}$ is on $l^p$ (where $\displaystyle l^p = \lbrace (x_1,x_2,...)| x_{i}\in\mathbb{C}\hspace{0.1cm}\text{and}\hspace{0.1cm} \left(\sum_{i=1}^{\infty}|x_{i}|^p\right)^{1/p}<\infty\rbrace$).

This is true for every $p>1$. Note that for $p=1$ this is the Harmonic series which is diverging. Let $f(x)= \frac{1}{x^p}$, then
$$\int_{1}^\infty f(x)dx =\lim_{b \to \infty}\int_{1}^b f(x)dx = \lim_{b \to \infty}\left[ \frac{x^{1-p}}{1-p} \right]_{1}^b = \frac{1}{p-1}$$
By the integral test for convergence it follows that the series
$$\sum_{k=1}^\infty f(k) = \sum_{k=1}^\infty \frac{1}{n^p} = < \infty$$
See here for more details about this series. 
