Binomial Theorem Approximations/Examples/Arbitrary Example 2

Example of Binomial Theorem Approximation

$\sqrt {25 \cdotp 1} \approx 5 \cdotp 0100$

to $4$ decimal places.

Proof

Although this article appears correct, it's inelegant. There has to be a better way of doing it.
In particular: Appallingly non-rigorous. Verbatim from the book.
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$\ds \sqrt {25 \cdotp 1}$

$=$

$\ds 5 \times \paren {1 + 0 \cdotp 004}^{1/2}$

$\ds $

$\approx$

$\ds 5 \times \paren {1 + \dfrac 1 2 \paren {0 \cdotp 004} + \dfrac {\paren {\frac 1 2} \paren {-\frac 1 2} } {2!} \paren {0 \cdotp 004}^2 + \cdots}$

$\ds $

$\approx$

$\ds 5 \times \paren {1 + 0 \cdotp 002 - 0 \cdotp 000002}$

$\ds $

$=$

$\ds 5 \cdotp 00999$

$\ds $

$\approx$

$\ds 5 \cdotp 0100$

$\blacksquare$

Sources

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: The Binomial Theorem: Approximations: Example $2$.