Prove $$f(x)=\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n}\chi_{[n-1,n)}(x)$$ is Riemann integrable
I don't understand this function so can you help me or give me some hint please?
2026-04-06 02:54:15.1775444055
Prove $f(x)=\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\chi_{[n-1,n)}(x)$ is Improper Riemann Integrable
85 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 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 INTEGRATION
- How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?
- How to integrate $\int_{0}^{t}{\frac{\cos u}{\cosh^2 u}du}$?
- Show that $x\longmapsto \int_{\mathbb R^n}\frac{f(y)}{|x-y|^{n-\alpha }}dy$ is integrable.
- How to find the unit tangent vector of a curve in R^3
- multiplying the integrands in an inequality of integrals with same limits
- Closed form of integration
- Proving smoothness for a sequence of functions.
- Random variables in integrals, how to analyze?
- derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound
- Which type of Riemann Sum is the most accurate?
Related Questions in IMPROPER-INTEGRALS
- multiplying the integrands in an inequality of integrals with same limits
- Closed form of integration
- prove that $\int_{-\infty}^{\infty} \frac{x^4}{1+x^8} dx= \frac{\pi}{\sqrt 2} \sin \frac{\pi}{8}$
- Generalized Fresnel Integration: $\int_{0}^ {\infty } \sin(x^n) dx $ and $\int_{0}^ {\infty } \cos(x^n) dx $
- Need a guide how to solve Trapezoidal rule with integrals
- For which values $p$ does $\int_0^\infty x\sin(x^p) dx $ converge?
- Proving $\int_0^1\frac{dx}{[ax+b(1-x)]^2}=\frac1{ab}$
- Contour integration with absolute value
- Use the comparison test to determine whether the integral is convergent or divergent.
- Can I simply integrate this function?
Related Questions in LEBESGUE-INTEGRAL
- A sequence of absolutely continuous functions whose derivatives converge to $0$ a.e
- Square Integrable Functions are Measurable?
- Lebesgue measure and limit of the integral.
- Solving an integral by using the Dominated Convergence Theorem.
- Convergence of a seqence under the integral sign
- If $g \in L^1$ and $f_n \to f$ a.e. where $|f_n| \leq 1$, then $g*f_n \to g*f$ uniformly on each compact set.
- Integral with Dirac measure.
- If $u \in \mathscr{L}^1(\lambda^n), v\in \mathscr{L}^\infty (\lambda^n)$, then $u \star v$ is bounded and continuous.
- Proof that $x \mapsto \int |u(x+y)-u(y)|^p \lambda^n(dy)$ is continuous
- a) Compute $T(1_{[\alpha,\beta]})$ for all $0<\alpha <\beta<0$
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?