Euler-Mascheroni constant

For $n$ a positive integer, let $H_{n}$ denote the $n^{\text{th}}$ partial sum of the harmonic series

Let $k > 1$ be an integer. Prove that

where $\log k$ is the natural logarithm of $k$, and $C$ is a constant.

Prove that the sequence

converges. Consequently, there is a formula

where $C=0.577216\ldots$ is called Euler's constant, and $\varepsilon_{n}\to 0$ as $n\to \infty$.

Find

From the inequalities $\left(1+\frac{1}{n}\right)^{n} < e < \left(1+\frac{1}{n}\right)^{n+1}$ it follows that

From this, prove the equality:

Here $C$ is a constant called Euler's constant, and $\varepsilon (n) \to 0$ as $n \to \infty$.

Using the previous result, prove the equality:

Prove Catalan's identity

and, using it, show that $\sum_{n=1}^{\infty } \frac{(-1)^{n-1}}{n} = \ln 2$.

Prove the equalities:

where $C$ is Euler's constant, and $\varepsilon_{n}$ and $\eta_{n} \rightarrow 0$ as $n \rightarrow \infty$.

Prove the equalities:

Having proved the convergence of the series $\sum_{n = 1}^{\infty }\left(\frac{1}{n}-\ln \frac{n+1}{n}\right)$, obtain the asymptotic formula $\sum_{k = 1}^{n} \frac{1}{k} = \ln n+C+\alpha_{n}$, where $C$ is Euler's constant $(C = 0,577215 \ldots ), \alpha_{n} \rightarrow 0$ as $n \rightarrow \infty$.

Using one of the equalities

where $\varepsilon_{n} \rightarrow 0$ as $n \rightarrow \infty$ (see problem 16$)$, prove that

Using the fact that the sum of the series $\sum_{n = 1}^{\infty } \frac{(-1)^{n-1}}{n}$ equals $\ln 2$, find the sums of the following series, obtained from the given one by rearranging its terms:

Prove that if the terms of the series $\sum_{n = 1}^{\infty } \frac{(-1)^{n}}{n}$ are rearranged so that each group of $p$ consecutive positive terms is followed by a group of $m$ consecutive negative terms, then the sum of the resulting series equals

Let the series $\sum_{n = 1}^{\infty } u_{n}$ be obtained from the series $\sum_{n = 1}^{\infty } \frac{(-1)^{n}}{n}$ by rearranging its terms so that the terms of like sign are arranged in the new series in decreasing order of their absolute values, and the ratio of the number of positive summands in $\sum_{k = 1}^{n} u_{k}$ to the number of negative summands in this sum has, as $n \rightarrow \infty$, a limit equal to $\lambda$. Prove that

Prove that $\prod_{n = 1}^{\infty } \frac{e^{1 / n}}{1+1 / n} = e^{C}$, where $C$ is Euler's constant:

Prove that

The number $\gamma =0.5772 \ldots$ is called Euler's constant.

Refine the results of the two previous problems by proving that for all $n \in \mathbb {N}$ the following inequalities hold:

$\sum_{n \geqslant 2}\left(\frac{1}{n} \sum_{m \geqslant 2} m^{-n}\right)$.

$\sum \frac{(-1)^{n}}{n} \ln n$.

$\sum \frac{(-1)^{n}}{n}\left[\log_{2} n\right]$.

$\sum \left(\frac{1}{2 n-1}-\ln \left(1+\frac{1}{2 n-1}\right)\right)$.

$\sum \left(\frac{1}{2 n}-\ln \left(1+\frac{1}{2 n}\right)\right)$.

Compute the integral $\int_{0}^{\infty } \frac{\ln x}{e^{x}} d x$.

Compute the integral $\int_{0}^{\infty } \frac{\ln^{2} x}{e^{x}} d x$.

Compute the integrals:

Compute the integrals:

Prove that as $p \rightarrow +0$

The expression

is bounded for increasing $n$; $C$ is Euler's constant [solution 18].

As it is well known [cf. E. T. Whittaker and G. N. Watson, p. 246] Euler's constant $C$ can be written as an integral

Show that

Let $\tau (n) = \sigma_{0}(n)$ denote the number of divisors of $n$. Then

$C$ is Euler's constant.

We may regard the relation

(Problem 18) as the definition of Euler's constant $C$. Derive hence that

The definition of $C$ given in Problem 19.1 is convenient. Yet it would be desirable to approximate $C$ by rational numbers, to represent $C$ as the sum of a series whose terms are rational. Prove that the following series fulfills this desideratum:

Take the terms in the order as you read a book: from left to right and from top to bottom. This is essential since the series is not absolutely convergent. Its terms are non increasing in absolute value.

$\uparrow$ Take $n_{0}=1, f_{7}=1$, and $G(x)=1 / x$, and derive $\sum_{n \leq x} n^{-1}=\log x+\gamma + O(1 / x)$, where $\gamma =1-\int_{1}^{\infty }(t-\lfloor t\rfloor ) t^{-2} d t$ is Euler's constant.