I was curious what does $$\cfrac{1}{1+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{\vdots}}}}$$ evaluate to. Empirically, I observed that it equals approximately $0.5819767$, and a calculator found that this value agrees with $\frac{1}{e-1}$ to at least 8 places. Is $\frac{1}{e-1}$ the exact value of this continued fraction? If this is true, is this result new? And how could the equivalence be proved?
2026-02-23 10:19:34.1771841974
What does $\frac{1}{1+\frac{2}{2+\frac{3}{{\vdots}}}}$ evaluate to?
276 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in REAL-ANALYSIS
- how is my proof on equinumerous sets
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Optimization - If the sum of objective functions are similar, will sum of argmax's be similar
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Justify an approximation of $\sum_{n=1}^\infty G_n/\binom{\frac{n}{2}+\frac{1}{2}}{\frac{n}{2}}$, where $G_n$ denotes the Gregory coefficients
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Is this relating to continuous functions conjecture correct?
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Absolutely continuous functions are dense in $L^1$
- A particular exercise on convergence of recursive sequence
Related Questions in CONTINUED-FRACTIONS
- Formula for the simple reapeted infinitely continued fractions
- Infinite continued fractions and convergents
- Convergence of $\ln\frac{x}{\ln\frac{x}{\ln x...}}$
- Find $\frac{a}{b} \in \mathbb{Q}$ such that $ |\,\frac{a}{b} - \sqrt{2}|_3 < \epsilon $
- $\sqrt{\frac{\pi e}{2}}=\frac{1}{1+\mathrm{K}_{i=1}^{\infty}{\frac{i}{1}}}+\sum_{n=0}^{\infty}{\frac{1}{(2n+1)!!}}$ implies $\sqrt{\pi e/2}\notin Q$?
- is there an algorithm that generates the continued fraction of a product of convergent continued fractions?
- continued fraction of $\sqrt{41}$
- Fundamental solution to specific Pell equation
- Continued fraction of binomial function $(1+z)^{1/4}$
- How does the convergence sector of a continued fraction depend on the order where it is truncated?
Related Questions in EULERS-NUMBER-E
- Non-Numerical proof of $e<\pi$
- Finding limit from Squeeze theorem: $\lim\limits_{n\to\infty} \left(\frac{2n-5}{3n+1}\right)^n$
- Limit of $a_n=(1- \frac{1}{n})^n$ as $n\rightarrow \infty$
- Why is Euler's number $2.71828$ and not anything else?
- What is the nature of $e^{ix}$, real or complex?
- Meaning of Maclaurin expansion of $e$
- Inequality involving $e$: $\binom{n}{k} < \frac{1}{e}\big(\frac{en}{k}\big)^k$
- weird properties of complex exponentials: $e^{i 2 \pi f(x)} = 1$
- Is $\left(1+\frac1n\right)^{n+1/2}$ decreasing?
- Intuitively, why are the two limit definitions of $e^x$ equivalent?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?