Convergence & divergence of geometric series

Convergence & divergence of geometric series

In this section, we will take a look at the convergence and divergence of geometric series. We've learned about geometric sequences in high school, but in this lesson we will formally introduce it as a series and determine if the series is divergent or convergent. For the first few questions we will determine the convergence of the series, and then find the sum. For the last few questions, we will determine the divergence of the geometric series, and show that the sum of the series is infinity.
Basic Concepts: Introduction to infinite series, Convergence and divergence of normal infinite series
Related Concepts: Arithmetic series

Lessons

Formulas for Geometric Series:

∑n=0∞arn=a1−r\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r} if -1 < rr < 1
∑n=1∞arn−1=a1−r\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{1-r} if -1 < rr < 1
If -1 < rr < 1, then the geometric series converges. Otherwise, the series diverges.
  • Introduction
    Geometric Series Overview:
  • 1.
    Convergence of Geometric Series
    Show that the following series are convergent and find its sum:
    a)
    ∑n=0∞13n\sum_{n=0}^{\infty} \frac{1}{3^n}

    b)
    ∑n=1∞[(−58)n−1+(1+3n7n)] \sum_{n=1}^{\infty} [(-\frac{5}{8})^{n-1}+(\frac{1+3^n}{7^n})]

    c)
    ∑n=0∞4n+223−4n\sum_{n=0}^{\infty}4^{n+2}2^{3-4n}

    d)
    ∑n=0∞42(n+2)53n−1\sum_{n=0}^{\infty} \frac{4^{2(n+2)}}{5^{3n-1}}

  • 2.
    Divergence of Geometric Series
    Show that the following series are divergent:
    a)
    ∑n=0∞3n−12n \sum_{n=0}^{\infty} \frac{3^{n-1}}{2^n}

    b)
    ∑n=0∞3n+223−n\sum_{n=0}^{\infty}3^{n+2}2^{3-n}

    c)
    ∑n=0∞[(14)n+(32)n2n]\sum_{n=0}^{\infty}[(\frac{1}{4})^n+(\frac{3}{2})^n2^n]

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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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