Abel's Lemma

The Abelian Lemma is a Lemma (mathematics) used to investigate the convergence series region of power series. It is named after Niels Henrik Abel.

Abel's Lemma

Let ${\displaystyle K_{P}:={z\in \mathbb {C} :P(z)=\sum _{k=0}^{\infty }a_{k}(z-z_{0})^{k}{\text{ converges}}}\subseteq \mathbb {C} }$ be the region of convergence of the power series ${\displaystyle P}$ given by: ${\displaystyle P(z)=\sum _{k=0}^{\infty }a_{k}(z-z_{0})^{k}}$, then the following statements hold:

• For a given element ${\displaystyle z_{1}\in K_{P}}$ from the convergence region of ${\displaystyle P}$, the series ${\displaystyle P(z)}$ converges absolutely for all ${\displaystyle z\in \mathbb {C} }$ such that ${\displaystyle |z-z_{0}|<|z_{1}-z_{0}|}$.
• For a given element ${\displaystyle z_{2}\notin K_{P}}$ where ${\displaystyle P}$ diverges, all ${\displaystyle z\in \mathbb {C} }$ with ${\displaystyle |z_{2}-z_{0}|<|z-z_{0}|}$ also cause ${\displaystyle P(z)}$ to diverge.

Task for Learners

• Prove the statement of Abel's Lemma by utilizing the fact that a convergent series (Absolute value) has bounded coefficients. Then, use the majorant criterion and a geometric series as a majorant to show that ${\displaystyle P}$ converges Absolute convergence.
• Justify why the convergence region ${\displaystyle K_{P}}$ contains an open disk ${\displaystyle D_{r}(z_{0})\subseteq {z\in \mathbb {C}$ :|z_{0}-z|<r}} (where ${\displaystyle r>0}$ is maximally chosen), and why ${\displaystyle P}$ diverges for all ${\displaystyle z\in \mathbb {C} }$ with ${\displaystyle r<|z-z_{0}|}$ when ${\displaystyle P}$ diverges.
• Determine the radius of convergence ${\displaystyle r>0}$ for the following power series, and on the boundary ${\displaystyle \partial D_{r}(0)}$ of the convergence region, identify two points ${\displaystyle z_{1},z_{2}\in \partial D_{r}(0)}$, such that ${\displaystyle P(z_{1})}$ converges and ${\displaystyle P(z_{2})}$ diverges. ${\displaystyle P(z)=\sum _{k=1}^{\infty }{\frac {1}{k}}\cdot z^{k}}$ Use your knowledge of the harmonic series to choose the points ${\displaystyle z_{1},z_{2}\in \partial D_{r}(0)}$.

(decomposition theorem) Analyze the decomposition theorem and explain how Abel's Lemma contributes to the extension of the domain to a ring and the use of the Identity Theorem.

Consequence

Taking into account that the series must always diverge at points ${\displaystyle z\in \mathbb {C} }$ where the sequence of its summands is unbounded (by the Cauchy's convergence test, it follows from the lemma that every power series has a well-defined radius of convergence and converges uniformly on any Compact space within the convergence disk. Outside the convergence disk, it diverges. No statement is made about the convergence for points on the boundary of the convergence disk.

See also

• Complex Analysis
• decomposition theorem

Source

Eberhard Freitag & Rolf Busam: Function Theory 1, Springer-Verlag, Berlin, ISBN 3-540-67641-4, p. 98

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