Analysis
Problems related to calculus, limits, series, etc.
Problems in this category
The problem is to determine the domain of the function $f(x) = \frac{1}{2\sqrt{x-3}}$.
DomainFunctionsSquare RootInequalities
2025/3/8
We are given a quadratic function $f(x) = x^2 - 2ax + 2$ defined on the interval $0 \le x \le 1$. We...
Quadratic FunctionsMinimum ValueIntervalVertexPiecewise Function
2025/3/8
The problem consists of two exercises. Exercise 1 deals with sequences. A sequence $U_n$ is defined ...
SequencesLimitsFunctionsDerivativesIntegralsAsymptotesGeometric SequencesExponential FunctionsLogarithmsArea Calculation
2025/3/7
We need to evaluate the definite integral $I = \int_{0}^{\infty} \frac{\sin(x)}{x} dx$. This is a cl...
Definite IntegralsImproper IntegralsDirichlet IntegralIntegration by PartsParameterizationDifferentiation under the integral sign
2025/3/7
We are asked to evaluate the indefinite integral of the function $\frac{x^2 + 72}{(x \sin x + 9 \cos...
IntegrationIndefinite IntegralTrigonometric Functions
2025/3/7
The problem is to find the sum of the infinite series $\sum_{k=1}^{\infty} \frac{1}{k^3}$.
Infinite SeriesRiemann Zeta FunctionApery's Constant
2025/3/7
The problem asks us to determine if the series $\sum_{k=1}^{\infty} k \sin \frac{1}{k}$ converges or...
Series ConvergenceLimit TestDivergence TestLimitsTrigonometric Functions
2025/3/7
The problem is to evaluate the infinite sum $\sum_{k=1}^{\infty} (\frac{1}{k} - \frac{1}{k+1})$.
Infinite SeriesTelescoping SeriesLimitsSummation
2025/3/7
The problem asks to evaluate the infinite sum $\sum_{k=1}^{\infty} \frac{1}{k^3}$. This is Apéry's c...
Riemann Zeta FunctionInfinite SeriesApery's ConstantNumber Theory
2025/3/7
We need to determine the convergence or divergence of the infinite series $\sum_{k=1}^{\infty} \frac...
Infinite SeriesConvergencep-seriesRiemann Zeta Function
2025/3/7