The harmonic series converges conditionally; therefore, the series can be rearranged in any way to get different sums, but how do we rearrange in such a way that it equates to 0. Is there a trick/ formula I could use to manipulate the series to equate it to 0 or to any number?
2026-02-23 00:30:15.1771806615
How do we rearrange the terms of the harmonic series so they add up to 0
289 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in SEQUENCES-AND-SERIES
- How to show that $k < m_1+2$?
- 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
- Negative Countdown
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Show that the sequence is bounded below 3
- A particular exercise on convergence of recursive sequence
- Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$
- Powers of a simple matrix and Catalan numbers
- Convergence of a rational sequence to a irrational limit
- studying the convergence of a series:
Related Questions in HARMONIC-NUMBERS
- A Gift Problem for the Year 2018
- Hypergeometric series with harmonic factor
- Infinite series with harmonic numbers related to elliptic integrals
- A non obvious example of a sequence $a(k)\cdot H_{b(k)}$ whose general term is integer many times, where $H_n$ denotes the $n$th harmonic number
- On integer sequences of the form $\sum_{n=1}^N (a(n))^2H_n^2,$ where $H_n$ is the $n$th harmonic number: refute my conjecture and add yourself example
- Simple formula for $H_n = m + \alpha $?
- Limit of the difference between two harmonic numbers
- Justify an approximation of $-\sum_{n=2}^\infty H_n\left(\frac{1}{\zeta(n)}-1\right)$, where $H_n$ denotes the $n$th harmonic number
- Show that for $n\gt 2$, $\frac{\sigma_1(n)}{n}\lt H_n$
- first derivative of exponential generating function of harmonic numbers
Related Questions in CONDITIONAL-CONVERGENCE
- Every non absolutely convergent series can be rearranged to converge to any $\lim \sup / \inf$ (Rudin)
- Conditional convergent improper Riemann integral vs. Lebesgue Integral
- Why does the commutative property of addition not hold for conditionally convergent series?
- Will the average of a conditional sample converge in probability to the conditional expectation?
- Conditional Convergent Power Series and radius of convergence
- How do we rearrange the terms of the harmonic series so they add up to 0
- $\sum_{n=0}^{\infty} \frac{(-1)^n}{1+\sqrt n}$
- When $\Big[ uv \Big]_{x\,:=\,0}^{x\,:=\,1}$ and $\int_{x\,:=\,0}^{x\,:=\,1} v\,du$ are infinite but $\int_{x\,:=\,0}^{x\,:=\,1}u\,dv$ is finite
- Is there an intuitive way of thinking why a rearranged conditionally convergent series yields different results?
- Conditional convergence of a series involving $sin n \theta$
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?
I let do the task by Mathematica, and obtained a surprise:
We see here not a more or less random sequence of positive and negative terms, depending on arithmetic accidents, but a definite pattern: There are blocks of five subsequent terms of the following kind: $${1\over 2n-1}, -{1\over 8n-6}, \ -{1\over 8n-4}, \ -{1\over 8n-2}, \ -{1\over 8n} \qquad(n\geq1)\ .$$ I hope that someone will be able to prove that the sequence will go on forever like that.