Using the ✍️integral test✍️ for convergence 🔥🔥 #apcalculus #apcalc #unit10 #shorts

We explore Topic 10.4 of AP Calculus BC, focusing on the integral test, which we can use to determine whether a series converges or diverges. This test requires that a series must be always positive, continuous, and decreasing.

Consider the series 1/n^3. Since the series starts at n=1, every term is positive, avoiding any potential discontinuity that would occur at n=0 due to a division by zero. By observing that the series has no discontinuities from n=1 onwards—all terms yield real number values—we confirm its continuity. Additionally, the series is always decreasing, as evidenced by the increasing denominator while the numerator remains constant, meeting all necessary conditions to apply the integral test.

The essence of the integral test lies in its direct relationship between the convergence of an integral and the series it represents. If the integral of the series from n=1 to infinity converges, so does the series; likewise, if the integral diverges, the series does too.

#APCalculusBC #IntegralTest #Convergence #Divergence #CalculusExplained #MathEducation

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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Видео Using the ✍️integral test✍️ for convergence 🔥🔥 #apcalculus #apcalc #unit10 #shorts канала Krista King