Binomial Theorem Approximations/Examples/Arbitrary Example 2

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Example of Binomial Theorem Approximation

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

to $4$ decimal places.


Proof



\(\ds \sqrt {25 \cdotp 1}\) \(=\) \(\ds 5 \times \paren {1 + 0 \cdotp 004}^{1/2}\)
\(\ds \) \(\approx\) \(\ds 5 \times \paren {1 + \dfrac 1 2 \paren {0 \cdotp 004} + \dfrac {\paren {\frac 1 2} \paren {-\frac 1 2} } {2!} \paren {0 \cdotp 004}^2 + \cdots}\)
\(\ds \) \(\approx\) \(\ds 5 \times \paren {1 + 0 \cdotp 002 - 0 \cdotp 000002}\)
\(\ds \) \(=\) \(\ds 5 \cdotp 00999\)
\(\ds \) \(\approx\) \(\ds 5 \cdotp 0100\)

$\blacksquare$


Sources