How fast does the sequence $(1+1/n)^n$ converge to $\mathrm{e}$?
Is the difference more like $\frac1n$ or more like $\mathrm{e}^{-n}$?
2026-03-25 20:42:18.1774471338
How fast does $\,\big(1+\frac 1n\big)^n$ converges to e?
277 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related 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 CALCULUS
- Equality of Mixed Partial Derivatives - Simple proof is Confusing
- How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?
- Proving the differentiability of the following function of two variables
- If $f ◦f$ is differentiable, then $f ◦f ◦f$ is differentiable
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Number of roots of the e
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Why the derivative of $T(\gamma(s))$ is $T$ if this composition is not a linear transformation?
- How to prove $\frac 10 \notin \mathbb R $
- Proving that: $||x|^{s/2}-|y|^{s/2}|\le 2|x-y|^{s/2}$
Related Questions in LIMITS
- How to prove $\lim_{n \rightarrow\infty} e^{-n}\sum_{k=0}^{n}\frac{n^k}{k!} = \frac{1}{2}$?
- limit points at infinity
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Maximal interval of existence of the IVP
- Divergence of power series at the edge
- Compute $\lim_{x\to 1^+} \lim_{n\to\infty}\frac{\ln(n!)}{n^x} $
- why can we expand an expandable function for infinite?
- Infinite surds on a number
- Show that f(x) = 2a + 3b is continuous where a and b are constants
- If $a_{1}>2$and $a_{n+1}=a_{n}^{2}-2$ then Find $\sum_{n=1}^{\infty}$ $\frac{1}{a_{1}a_{2}......a_{n}}$
Related Questions in CONVERGENCE-DIVERGENCE
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Conditions for the convergence of :$\cos\left( \sum_{n\geq0}{a_n}x^n\right)$
- Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$
- Pointwise and uniform convergence of function series $f_n = x^n$
- studying the convergence of a series:
- Convergence in measure preserves measurability
- If $a_{1}>2$and $a_{n+1}=a_{n}^{2}-2$ then Find $\sum_{n=1}^{\infty}$ $\frac{1}{a_{1}a_{2}......a_{n}}$
- Convergence radius of power series can be derived from root and ratio test.
- Does this sequence converge? And if so to what?
- Seeking an example of Schwartz function $f$ such that $ \int_{\bf R}\left|\frac{f(x-y)}{y}\right|\ dy=\infty$
Related Questions in RATE-OF-CONVERGENCE
- sublinear rate of convergence in mathematical optimization
- Estimate rate of convergence for a sequence to a limit
- Estimate convergence rate for recurrences $a_{k} \le \frac{k}{k+2} a_{k-1}$ and $b_{k} \le \frac{k+\alpha}{k+2} b_{k-1}$
- Convergence rate if a sequence $\{x_k\}$ satisfies that $x_{k} - x_{k-1} \le \frac {1} {k^{p}}$ where $p >1$
- Estimate convergence order of a sequence
- Rate of Convergence of Function F(x) = f(x)/f′(x) using Newtons Method
- Speed of convergence of an integral (whose complete version gives the Mascheroni constant)
- Convergence of a linear recurrence equation
- Convergence rate of $\operatorname E|\langle X,f_n\rangle|^p$
- is there a better way to prove it?
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?
Observe that $$ \left(1+\frac{1}{n}\right)^n=\mathrm{e}^{n\log(1+\frac{1}{n})} =\mathrm{e}^{n(\frac{1}{n}-\frac{1}{2n^2}+\mathcal O(n^{-3}))} =\mathrm{e}\cdot\mathrm{e}^{-\frac{1}{2n}+\mathcal O(n^{-2}))}=\mathrm{e}\left(1-\frac{1}{2n}+\mathcal O(n^{-2})\right) $$ Hence $$ \left(1+\frac{1}{n}\right)^n-\mathrm{e}=-\frac{\mathrm{e}}{2n}+\mathcal O(n^{-2}) $$