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:

\(\displaystyle \tan x\) \(=\) \(\displaystyle \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} x^{2 n - 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {2^2 \paren {2^2 - 1} B_2} {2!} x - \frac {2^4 \paren {2^4 - 1} B_4} {4!} x^3 + \frac {2^6 \paren {2^5 - 1} B_6} {6!} x^5 - \frac {2^8 \paren {2^7 - 1} B_8} {8!} + \frac {2^{10} \paren {2^9 - 1} B_{10} } {10!} - \cdots\)


Enumerating the Bernoulli numbers:

\(\displaystyle B_2\) \(=\) \(\displaystyle \dfrac 1 6\)
\(\displaystyle B_4\) \(=\) \(\displaystyle -\dfrac 1 {30}\)
\(\displaystyle B_6\) \(=\) \(\displaystyle \dfrac 1 {42}\)
\(\displaystyle B_8\) \(=\) \(\displaystyle -\dfrac 1 {30}\)
\(\displaystyle B_{10}\) \(=\) \(\displaystyle \dfrac 5 {66}\)


Thus the appropriate arithmetic is performed on each coefficient:


\(\displaystyle \frac {2^2 \paren {2^2 - 1} B_2} {2!}\) \(=\) \(\displaystyle \frac {4 \times 3} {2} \times \dfrac 1 6\)
\(\displaystyle \) \(=\) \(\displaystyle 1\)


\(\displaystyle -\frac {2^4 \paren {2^4 - 1} B_4} {4!}\) \(=\) \(\displaystyle -\dfrac {16 \times 15} {24} \times \dfrac {-1} {30}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\paren {2^4} \times \paren {3 \times 5} } {\paren {2^3 \times 3} \times \paren {2 \times 3 \times 5} }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 3\)


\(\displaystyle \frac {2^6 \paren {2^5 - 1} B_6} {6!}\) \(=\) \(\displaystyle \dfrac {64 \times 63} {720} \times \dfrac 1 {42}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\paren {2^6} \times \paren {3^2 \times 7} } {\paren {2^4 \times 3^2 \times 5} \times \paren {2 \times 3 \times 7} }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 2 {15}\)


\(\displaystyle -\frac {2^8 \paren {2^7 - 1} B_8} {8!}\) \(=\) \(\displaystyle -\dfrac {256 \times 255} {40 \, 320} \times \dfrac {-1} {30}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\paren {2^8} \times \paren {3 \times 5 \times 17} } {\paren {2^7 \times 3^2 \times 5 \times 7} \times \paren {2 \times 3 \times 5} }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {17} {3^2 \times 5 \times 7}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {17} {315}\)


\(\displaystyle \frac {2^{10} \paren {2^9 - 1} B_{10} } {10!}\) \(=\) \(\displaystyle \dfrac {1024 \times 1023} {3 \, 628 \, 800} \times \dfrac 5 {66}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\paren {2^{10} } \times \paren {3 \times 11 \times 31} \times 5} {\paren {2^8 \times 3^4 \times 5^2 \times 7} \times \paren {2 \times 3 \times 11} }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {2 \times 31} {\paren {3^4 \times 5 \times 7} }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {62} {2835}\)

Hence the result.

$\blacksquare$


Sources

Beware: there is a mistake in the $5$th term as reported here.