Integral test

Integral test

In this section, we will learn about another test called the Integral test. The idea is to take the general term as a function in terms of x, and then integrate it. You can only use this test if the function is positively decreasing. If the integral gives a finite value, then the series is convergent. If the integral diverges to infinity, then the series is also divergent. We will first do some questions that require you to use the integral test instead of p-series test. Then we will use the integral test on a complicated series to see if it converges or diverges.
Basic Concepts: Improper integrals, Introduction to infinite series

Lessons

Note *The integral test states the following:
If f(x)=anf(x)=a_n and f(x)f(x) is a continuous, positive decreasing function from [i,∞][i,\infty], then we can say that:
1. If ∫i∞f(x)dx\int_{i}^{\infty}f(x)dx is convergent, then the series ∑n=i∞an\sum_{n=i}^{\infty}a_n is also convergent.
2. If ∫i∞f(x)dx\int_{i}^{\infty}f(x)dx is divergent, then the series ∑n=i∞an\sum_{n=i}^{\infty}a_n is also divergent.
  • Introduction
    Integral Test Overview
  • 1.
    P Series versus Integral test
    Use the integral test instead of the p-series test to show that the series converge or diverge.
    a)
    ∑n=1∞3n2\sum_{n=1}^{\infty}\frac{3}{n^2}

    b)
    ∑n=1∞1n\sum_{n=1}^{\infty}\frac{1}{n}

  • 2.
    Convergence/Divergence of Integral Test
    Determine whether the following series converge or diverge using the integral test.
    a)
    ∑n=3∞2(5n+4)5\sum_{n=3}^{\infty}\frac{2}{(5n+4)^5}

    b)
    ∑n=1∞1n2+7n+12\sum_{n=1}^{\infty}\frac{1}{n^2+7n+12}

  • 3.
    Advanced Question Regarding to the Integral Test
    Determine if the series ∑k=2∞1k  3ln(4k)\sum_{k=2}^{\infty}\frac{1}{k\ \ {^3}\sqrt{ln(4k)}} converges or diverges.
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8.
Sequence and Series
8.1
Introduction to sequences
8.2
Introduction to infinite series
8.3
Convergence and divergence of normal infinite series
8.4
Convergence and divergence of geometric series
8.5
Divergence of harmonic series
8.6
P Series
8.7
Alternating series test
8.8
Divergence test
8.9
Comparison and limit comparison test
8.10
Integral test
8.11
Ratio test
8.12
Absolute and conditional convergence
8.13
Radius and interval of convergence with power series
8.14
Functions expressed as power series
8.15
Taylor and maclaurin series
8.16
Approximating functions with Taylor polynomials and error bounds
AP Calculus BC
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