Exercise 6.5.11

Question

A series $\sum_{n=0}^{\infty} a_{n}$ is said to be Abel-summable to $L$ if the power series
$$
f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}
$$
converges for all $x \in[0,1)$ and $L=\lim _{x \rightarrow 1-} f(x)$.

\begin{enumerate}[label=(\alph*)] 
\item Show that any series that converges to a limit $L$ is also Abel-summable to $L$.
\item Show that $\sum_{n=0}^{\infty}(-1)^{n}$ is Abel-summable and find the sum.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item If a series $\sum^\infty_{n=0}a_n$ converges to $L$ then by Abel's Theorem the power series $\sum^\infty_{n=0}a_n x^n$ converges uniformly over $[0,1]$, and is therefore continuous over this interval. Hence by continuity the series is Abel-summable to $L$.
\item The relevant power series here is $f(x) = \sum^\infty_{n=0} (-x)^n$ which has the closed-form expression $\frac{1}{1 + x}$ for $\left|x\right| < 1$, and $\lim_{x \to 1^-} f(x)$ evaluates to $1/2$.
 \end{enumerate}

Exercise 6.5.11

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Exercise 6.5.11

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