7. Power Series

Inside the radius of convergence — Abel and Tauber theorems, rearrangements, Cauchy products.

• Power Series and Radius of Convergence — A power series $\sum a_n x^n$ converges absolutely inside its radius $R$ and diverges outside — watch three archetypal series pin down their radii, and see why the boundary $\lvert x\rvert = R$ must be decided case by case.
• Hadamard Gap Theorem — $S_N(z) = \sum_{k=0}^{N} z^{2^k}$ (gap ratio exactly 2): plotting $|S_N(r e^{i\theta})|$ for $r \in \{0.90, 0.95, 0.99\}$, spikes erupt at more and more angles as $N$ grows — a gappy series with radius exactly 1 is singular at every point of the unit circle. The gap-free contrast $\sum z^k/(k+1)^2$ stays calm.
• Abel's Theorem — For a convergent series $\sum a_n \to S$, the power series $\sum a_n x^n$ approaches $S$ as $x \to 1^{-}$ — Abel's theorem. Even the divergent Grandi series points to $\frac{1}{2}$ in this limit, but that is the Abel sum, never "the sum" of the series.
• Tauber's Theorem — Grandi's series has Abel sum $\frac{1}{2}$ yet diverges — because $|n\,a_n| = n$ breaks the hypothesis $a_n = o(1/n)$. The series $d_n = \frac{(-1)^n}{(n+1)\ln(n+2)}$ satisfies it, and its Abel-sum curve reaches the ordinary sum — Tauber's theorem.
• Two Roads to e — Compare how fast compound growth $(1+\frac{1}{n})^n$ and the series $\sum_{k=0}^{\infty} \frac{1}{k!}$ approach $e$. The factorial series wins by a landslide.
• ζ(4) and Even Powers — The partial sums of $\frac{1}{n^4}$ reach $\frac{\pi^4}{90}$ quickly — and at every even power, the $\zeta$ value closes into a power of $\pi$.
• Rearranging Changes the Sum — Reordering a conditionally convergent series changes its sum — $\ln 2 \approx 0.6931$ becomes $\frac{3}{2}\ln 2 \approx 1.0397$. Twin of the Leibniz scene.
• Absolute Convergence — If the absolute-value series converges, so does the original — compare the two partial-sum walks $S_n$ and $T_n$ settling together against the harmonic-like case where the signs rescue a divergent absolute series.
• Riemann's Rearrangement Theorem — A conditionally convergent series can be rearranged to sum to any real number — watch the greedy block rule steer the alternating harmonic series (which sums to $\ln 2$ in its natural order) onto a target of your choice.
• The Cauchy Product — $(\sum a_n)(\sum b_n) = \sum c_n$ with $c_n = \sum_{k=0}^{n} a_k b_{n-k}$ — sweeping the lattice along diagonals. Mertens' theorem (absolute convergence on one side saves the product) and the counterexample when both converge only conditionally.