Expand 0.444… as
0.444… = 4 × 0.111…
0.444… = 4 × (0.1 + 0.01 + 0.001 + …)
0.444… = 4 × (1/10 + 1/100 + 1/1000 + …)
0.444… = 4 × (1/10¹ + 1/10² + 1/10³ + …)
so that
[tex]0.444\ldots = 4 \displaystyle \sum_{n=1}^\infty \frac1{10^n}[/tex]
