😳😳 How to use the INTEGRAL TEST for convergence 🥸🥸 #apcalculus #apcalc #unit5 #shorts

The integral test for convergence is a critical concept in calculus that assesses whether a series converges. To apply this test to a series, the series must be positive, continuous, and decreasing. We transform the series into a function by swapping n for x, enabling us to integrate the series as a function. Given the nature of this test involving an improper integral due to the upper limit stretching to infinity, we substitute infinity with a variable b, and then explore the limit as b approaches infinity.

We integrate, evaluate the integral over our interval, and then evaluate the limit. In our example, as b marches towards infinity, the first term fades to 0, leaving us with a finite value of 1/2, signaling that the integral converges. The conclusion of the integral test is this: the convergence of the integral directly implies the convergence of the series it represents.

#IntegralTest #Convergence #Calculus #MathTutorial #SeriesConvergence

Unit 10 of AP Calculus is all about Infinite Sequences and Series:
10.1 Defining Convergent and Divergent Infinite Series
10.2 Working with Geometric Series
10.3 The nth Term Test for Divergence
10.4 Integral Test for Convergence
10.5 Harmonic Series and p-Series
10.6 Comparison Tests for Convergence
10.7 Alternating Series Test for Convergence
10.8 Ratio Test for Convergence
10.9 Determining Absolute or Conditional Convergence
10.10 Alternating Series Error Bound
10.11 Finding Taylor Polynomial Approximations of Functions
10.12 Lagrange Error Bound
10.13 Radius and Interval of Convergence of Power Series
10.14 Finding Taylor or Maclaurin Series for a Function
10.15 Representing Functions as Power Series

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[offthewallreubs]

the integral test blew my mind when I first learnt it. I think it's really neat ! :)

[abcturin]

Harmonic series: E 1/n^q, if q>1, conv. If q<=1, div.

[MathisLife-ih]

Integral test, as long as it does exist
then we can say that the series is convergent

[AliHassan-hbbn]

I don't like the term approches +_ infinity. Better say goes to +- infinity. However I would say approaches to anumber. No one can approach infinity.

[AliHassan-hbbn]

When the bigger cousin converges so does the younger.