Binomial Theorem/Examples/Square Root of 2
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Example of Use of Binomial Theorem
- $\sqrt 2 = 2 \paren {1 - \dfrac 1 {2^2} - \dfrac 1 {2^5} - \dfrac 1 {2^7} - \dfrac 5 {2^{11} } - \cdots}$
Proof
| \(\ds \sqrt 2\) | \(=\) | \(\ds 2 \times \frac 1 2 \times \sqrt 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \times \sqrt {\frac 1 4 } \times \sqrt 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \times \sqrt {\frac 1 2 }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \paren {1 - \frac 1 2}^ \frac 1 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \paren {1 + \paren { \frac 1 2 } \paren { -\frac 1 2 } + \dfrac {\paren { \frac 1 2 } \paren {\paren { \frac 1 2 } - 1} } {2!} \paren { -\frac 1 2 }^2 + \dfrac {\paren { \frac 1 2 } \paren {\paren { \frac 1 2 } - 1} \paren {\paren { \frac 1 2 } - 2} } {3!} \paren { -\frac 1 2 }^3 + \dfrac {\paren { \frac 1 2 } \paren {\paren { \frac 1 2 } - 1} \paren {\paren { \frac 1 2 } - 2} \paren {\paren { \frac 1 2 } - 3} } {4!} \paren { -\frac 1 2 }^4 + \cdots}\) | General Binomial Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \paren {1 - \dfrac 1 {2^2} - \dfrac 1 {2^5} - \dfrac 1 {2^7} - \dfrac 5 {2^{11} } - \cdots}\) |
The first few terms of the real sequence are:
- $2, \dfrac 3 2, \dfrac {23} {16}, \dfrac {91} {64}, \dfrac {1451} {1024}, \dotsc$
$\blacksquare$