34. Euler's Constant and Rearranging Series

Euler's Constant

Let

$${\gamma }_n=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}-\log n=\sum _{k=1}^n\frac{1}{k}-{\int }_1^n\frac{1}{x}dx$$

By Integral Test using $f:[1,\infty )\to \mathbb{R}$ by $f(x)=\frac{1}{x}$ then $({\gamma }_n)$ converges
Suppose

$${\gamma }_n\to \gamma \text{ as }n\to \infty$$

and $0\le \gamma \le 1$ (from the test)

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Rearrangement of Series

Let $g:\mathbb{N}\to \mathbb{N}$ be a bijection
Given a series $\sum a_k$
Let $b_k=a_{g(k)}$
Then

$$\sum b_k\text{ is a rearrangement of }\sum a_k$$

Note that if $\sum a_k$ is absolutely convergent then any rearrangement yields the same limit

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