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Calculus 2

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All Calculus 2Volumes of Solids of RevolutionIntegration by PartsTrigonometric IntegralsTrigonometric substitutionPartial fractionsImproper integralsStrategy for integrationArc lengthArea of a surface of revolutionIntroduction to differential equationsSeparable differential equationsLinear differential equationsParametrized curvesPolar coordinatesSequencesSeries and the integral testComparison testsAlternating series and absolute convergenceRatio and root testsPower series and representations of functionsTaylor and Maclaurin seriesApplications of Taylor polynomials
Using the Integral Test on Exponential Series

Try using the integral test on your own for the series ∑n=1∞en\sum_{n=1}^{\infty} e^n and determine if it converges or diverges.

Series and the integral testMediumVideo
Improper Integrals and Series Convergence

If you look at the improper interval and the improper integral converges or diverges, whatever it does, the same is true of the series.

Improper integralsMediumVideoNeeds Attention
Convergence of Series Using the Integral Test

Calculate whether the series ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2} is convergent using the integral test, and estimate its sum.

Series and the integral testMediumVideo
Convergence of Series with Comparison Test

Using the comparison test, determine if the series ∑n=1∞12n+1\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^n + 1} is convergent by comparing it to the series ∑n=1∞12n\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^n}.

Comparison testsMediumVideo
Convergence of Series Using Comparison Test

Determine whether the series ∑n=1∞52n2+4n+3\displaystyle \sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3} is convergent using the comparison test.

Comparison testsMediumVideo
Series Convergence with Integral Test

Determine whether the series represented by the function is convergent or divergent using the integral test for the function: ∫1∞12x dx\displaystyle \int_{1}^{\infty} \frac{1}{2^x} \, dx

Series and the integral testMediumVideo
Convergence or Divergence of Series Using Integral Test

Determine whether the series represented by the function is convergent or divergent using the integral test for the function: ∫1∞xx2+1 dx\displaystyle \int_{1}^{\infty} \frac{x}{x^2 + 1} \, dx

Series and the integral testMediumVideo
Convergence of the Alternating Harmonic Series

Consider the series that starts from one goes to infinity: (−1)n⋅1n(-1)^n \cdot \frac{1}{n}. Will the alternating harmonic series converge or diverge?

Alternating series and absolute convergenceMediumVideo
Convergence of Alternating Series with Rational Function Term

Consider the series which goes from 1 to infinity: (−1)n+1⋅5n+32n−7(-1)^{n+1} \cdot \frac{5n + 3}{2n - 7}. Will this series converge or diverge?

Alternating series and absolute convergenceMediumVideo
Convergence of an Alternating Series

Consider the series: (−1)n+15n\frac{(-1)^{n+1}}{5^n}. Will the series converge or diverge?

Alternating series and absolute convergenceMediumVideo
Convergence of Alternating Series with Logarithmic Denominator

Consider the series (−1)n⋅nln⁡(n)(-1)^n \cdot \frac{n}{\ln(n)}. Will the series converge or diverge?

Alternating series and absolute convergenceMediumVideo
Alternating Series Convergence of Logarithmic Over n

Let's say the series is (−1)n+1⋅ln⁡(n)n(-1)^{n+1} \cdot \frac{\ln(n)}{n}. Will it converge or diverge?

Alternating series and absolute convergenceMediumVideo
Alternating Series Test for Cosine Series

Consider the series cos⁡(nπ)n\frac{\cos(n \pi)}{n}. Can we apply the alternating series test to it?

Alternating series and absolute convergenceMediumVideo
Convergence of Alternating Series

Take the series (−1)n−1Bn(-1)^{n-1} B_n, where BnB_n is positive. Determine if the series is convergent or divergent based on if Bn+1≤BnB_{n+1} \leq B_n and lim⁡n→∞Bn=0\lim_{{n \to \infty}} B_n = 0.

Alternating series and absolute convergenceMediumVideoNeeds Attention
Alternating Series Convergence Test for Series Involving Fractions

Apply the alternating series test to different series to determine convergence or divergence: (−1)n⋅3n−12n+1(-1)^n \cdot \frac{3n-1}{2n+1} and (−1)n+1⋅n2n3+4(-1)^{n+1} \cdot \frac{n^2}{n^3+4}.

Alternating series and absolute convergenceMediumVideoNeeds Attention
Convergence of n to the n over n Factorial Using Ratio Test

Use the ratio test to determine the convergence of the series nnn!\frac{n^n}{n!}.

Ratio and root testsMediumVideoNeeds Attention
Convergence of Alternating Harmonic Series Using Alternating Series Test

Given an alternating series ∑(−1)n−11n\sum (-1)^{n-1} \frac{1}{n}, determine if the series converges using the Alternating Series Test.

Alternating series and absolute convergenceMediumVideo
Convergence of Alternating Series2

Determine if the series ∑n=1∞(−1)n−14n−1\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{4n-1} converges using the alternating series test.

Alternating series and absolute convergenceMediumVideo
Convergence of Series with Rational Function

Determine if the series ∑n=1∞n+13n+2\sum_{n=1}^{\infty} \frac{n+1}{3n+2} converges or diverges, and justify your answer.

Comparison testsMediumVideo
Alternating Series Test for Convergence

Determine if the series ∑n=1∞nn2+1\sum_{n=1}^{\infty} \frac{n}{n^2+1} converges using the alternating series test.

Series and the integral testMediumVideo