Write the first four non-zero terms of the power series

Write the first four non-zero terms of the power series in $x$ for $(1+x)^x$.

Step 1 · Rewrite and expand $u$

$$ (1+x)^x = e^{x\ln(1+x)}, \qquad u=x\ln(1+x)=x^2-\frac{x^3}2+\frac{x^4}3-\frac{x^5}4+\cdots $$

Step 2 · Powers of $u$

$$ u^2 = x^4-x^5+\frac{11x^6}{12}+\cdots, \qquad u^3 = x^6+\cdots $$

Step 3 · Collect $e^u=1+u+\frac{u^2}2+\frac{u^3}6+\cdots$

Coefficient of $x^2$: $1$. Coefficient of $x^3$: $-\frac12$. Coefficient of $x^4$: $\frac13+\frac12=\frac56$.

Summary Table

Additional Notes

• There is no linear term: the series begins $1+x^2+\cdots$ because $x\ln(1+x)\sim x^2$ near $0$, so composing with $e^u$ delays the first nontrivial correction to second order.
• This variable-exponent construction — writing $f(x)^{g(x)}=e^{g(x)\ln f(x)}$ before expanding — is the standard technique for any power tower where both base and exponent depend on $x$, and is the same idea used for $a^x$ in Problem 254 (there with constant base instead).
• The radius of convergence, $1$, is inherited from $\ln(1+x)$'s singularity at $x=-1$, since $e^u$ is entire and cannot introduce any new restriction.