Long meme Infinite sum

$\begin{aligned} x & = \frac {69}{1\cdot 420} + \frac {69}{1\cdot 421} + \frac {69}{1\cdot 422} + \cdots + \blue{\frac {69}{n(419+n)}} + \cdots \\ & = \sum_{n=1}^\infty \blue{\frac {69}{n(419+n)}} & \small \blue{\text{By partial fraction decomposition}} \\ & = \frac {69}{419} \sum_{n=1}^\infty \left(\frac 1n - \frac 1{n+419} \right) \\ & = \frac {69}{419} \left(\sum_{n=1}^\infty \frac 1n - \sum_{n=420}^\infty \frac 1n \right) \\ & = \frac {69}{419} \blue{\left(\frac 11+\frac 12 + \frac 13 + \cdots + \frac 1{419}\right)} \\ & = \frac {69}{419} \blue{H(419)} & \small \blue{H(n) \text{ denotes the }n\text{th harmonic number.}} \\ & \approx \frac {69}{419} \cdot \blue{6.616279428} & \small \blue{\text{By Excel spreadsheet}} \\ & \approx 1.04 \end{aligned}$

Therefore, $\boxed{1<x<1.1}$ .

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Reference: Harmonic number