Power Series Expansion for Tangent Function/Sequence
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Theorem
The Power Series Expansion for Tangent Function begins:
- $\tan x = x + \dfrac 1 3 x^3 + \dfrac 2 {15} x^5 + \dfrac {17} {315} x^7 + \dfrac {62} {2835} x^9 + \cdots$
The numerators form sequence A002430 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
The denominators form sequence A036279 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
From Power Series Expansion for Tangent Function:
\(\ds \tan x\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} x^{2 n - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2^2 \paren {2^2 - 1} B_2} {2!} x - \frac {2^4 \paren {2^4 - 1} B_4} {4!} x^3 + \frac {2^6 \paren {2^6 - 1} B_6} {6!} x^5 - \frac {2^8 \paren {2^8 - 1} B_8} {8!} x^7 + \frac {2^{10} \paren {2^{10} - 1} B_{10} } {10!} x^9 - \cdots\) |
Enumerating the Bernoulli numbers:
\(\ds B_2\) | \(=\) | \(\ds \dfrac 1 6\) | ||||||||||||
\(\ds B_4\) | \(=\) | \(\ds -\dfrac 1 {30}\) | ||||||||||||
\(\ds B_6\) | \(=\) | \(\ds \dfrac 1 {42}\) | ||||||||||||
\(\ds B_8\) | \(=\) | \(\ds -\dfrac 1 {30}\) | ||||||||||||
\(\ds B_{10}\) | \(=\) | \(\ds \dfrac 5 {66}\) |
Thus the appropriate arithmetic is performed on each coefficient:
\(\ds \frac {2^2 \paren {2^2 - 1} B_2} {2!}\) | \(=\) | \(\ds \frac {4 \times 3} {2} \times \dfrac 1 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
\(\ds -\frac {2^4 \paren {2^4 - 1} B_4} {4!}\) | \(=\) | \(\ds -\dfrac {16 \times 15} {24} \times \dfrac {-1} {30}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {2^4} \times \paren {3 \times 5} } {\paren {2^3 \times 3} \times \paren {2 \times 3 \times 5} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 3\) |
\(\ds \frac {2^6 \paren {2^6 - 1} B_6} {6!}\) | \(=\) | \(\ds \dfrac {64 \times 63} {720} \times \dfrac 1 {42}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {2^6} \times \paren {3^2 \times 7} } {\paren {2^4 \times 3^2 \times 5} \times \paren {2 \times 3 \times 7} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 2 {15}\) |
\(\ds -\frac {2^8 \paren {2^8 - 1} B_8} {8!}\) | \(=\) | \(\ds -\dfrac {256 \times 255} {40 \, 320} \times \dfrac {-1} {30}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {2^8} \times \paren {3 \times 5 \times 17} } {\paren {2^7 \times 3^2 \times 5 \times 7} \times \paren {2 \times 3 \times 5} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {17} {3^2 \times 5 \times 7}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {17} {315}\) |
\(\ds \frac {2^{10} \paren {2^{10} - 1} B_{10} } {10!}\) | \(=\) | \(\ds \dfrac {1024 \times 1023} {3 \, 628 \, 800} \times \dfrac 5 {66}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {2^{10} } \times \paren {3 \times 11 \times 31} \times 5} {\paren {2^8 \times 3^4 \times 5^2 \times 7} \times \paren {2 \times 3 \times 11} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 \times 31} {\paren {3^4 \times 5 \times 7} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {62} {2835}\) |
Hence the result.
$\blacksquare$
Sources
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler
- Beware: there is a mistake in the $5$th term as reported here.