Expand (1+x²) arctan x in a power series in x
Expand $(1+x^2)\arctan x$ in a power series in $x$.
Explanation of Method
Multiplying the $\arctan x$ series by $(1+x^2)$ splits into the original series plus a copy shifted up by two powers; reindexing and combining the two odd-power series with a partial-fraction-style algebraic simplification yields a single closed-form coefficient.
- Multiply the $\arctan x$ series by $1$ and by $x^2$ separately.
- Reindex the $x^2$-shifted sum and combine like powers.
- Simplify the combined coefficient and verify at $x=1$.
Step 1 · Multiply term by term
With $\arctan x=\sum_{n=0}^{\infty}c_nx^{2n+1}$, $c_n=\frac{(-1)^n}{2n+1}$: $$ (1+x^2)\arctan x = \sum_{n=0}^{\infty}c_nx^{2n+1} + \sum_{n=0}^{\infty}c_nx^{2n+3}. $$
Step 2 · Reindex and combine
In the second sum let $m=n+1$: $\sum_{m=1}^{\infty}c_{m-1}x^{2m+1}$. The $n=0$ term of the first sum, $c_0x=x$, has no partner; for $n\ge1$: $$ (c_n+c_{n-1})x^{2n+1}, \qquad c_n+c_{n-1} = (-1)^n\left(\frac1{2n+1}-\frac1{2n-1}\right) = \frac{2(-1)^{n+1}}{4n^2-1}. $$
Step 3 · Final series and check
$$ (1+x^2)\arctan x = x + \sum_{n=1}^{\infty}\frac{2(-1)^{n+1}}{4n^2-1}x^{2n+1} = x+\frac{2x^3}3-\frac{2x^5}{15}+\frac{2x^7}{35}-\cdots $$ At $x=1$: $1+\frac23-\frac2{15}+\frac2{35}-\cdots\to\frac\pi2=2\arctan(1)\cdot1$, i.e. $(1+1)\arctan1=2\cdot\frac\pi4=\frac\pi2$, matching numerically.
Summary Table
| Property | Value |
|---|---|
| Series | $x+\sum_{n\ge1}\frac{2(-1)^{n+1}}{4n^2-1}x^{2n+1}$ |
| Denominator | $4n^2-1=(2n-1)(2n+1)$ |
| Region of validity | $|x|\le1$ |
Additional Notes
- The reindexing trick used here — splitting off the unmatched lowest-order term before combining the two shifted sums — is a standard technique whenever a power series is multiplied by a polynomial in $x^2$.
- The convergence region is unchanged from that of the base $\arctan x$ series (Problem 267), since multiplying by the polynomial $1+x^2$ can never shrink a radius of convergence.
- Differentiating gives $\frac{d}{dx}[(1+x^2)\arctan x]=2x\arctan x+1$, which can be independently verified by differentiating the combined series term by term.