Analysis
Problems related to calculus, limits, series, etc.
Problems in this category
We are asked to find the value of the infinite sum $\sum_{k=1}^{\infty} \frac{1000k^2}{1+k^2}$.
Infinite SeriesDivergence TestLimits
2025/3/9
The problem presents two functions, $g(x)$ and $f(x)$. $g(x) = -(x-1)^2 + 1 - \ln(x-1)$ $f(x) = -x +...
FunctionsLimitsDerivativesAsymptotesBijective FunctionsInverse Functions
2025/3/8
The problem consists of two exercises. Exercise 1 deals with sequences and their convergence. Specif...
SequencesConvergenceMatricesMatrix InverseGaussian EliminationLinear EquationsGeometric Series
2025/3/8
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