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    • ▾Calculus 2
      • ▸Integration techniques
        • •Intro to calc 2
        • ▸Basic indefinite integrals
          • •Reverse power rule for indefinite integrals
          • •Sum, difference, and reverse power rules for indefinite integrals
        • •Integration by parts
        • ▸Integrals involving trig functions
          • •Lesson
          • •Practice
        • •Trig substitutions
        • •Integration by partial fractions
        • •Integral recurrence relations
        • •Integrating quadratics by completing the square
        • •Integration by long division
        • •Improper integrals
        • •Comparison tests for convergence
        • ▸Approximating definite integrals
          • •Riemann sums
          • ▸Simpson's rule
            • •Practice: Simpson's rule
          • ▸Trapezoidal rule
            • •Lesson
            • •Practice
          • •Midpoint rule
      • ▸Applications of integrals
        • ▸Arc length (calculus 2)
          • •Using the formula
        • ▸Center of mass (calculus 2)
          • •Center of mass for one and two-dimensional systems
          • •Center of mass for planar lamina
          • •Pappus's theorem
        • •Hydrostatic force
        • •Mode of a continuous random variable from a probability density function
        • •Surface area of solids of revolution
        • •Gabriel's Horn
        • •Applications of trigonometric integrals
        • •Integrating exponential functions
        • •Logarithmic functions
      • ▸Parametric equations and polar coordinates
        • •Cycloid area and length
        • •Eliminating the parameter
        • •Testing polar equations for symmetry
        • •Deriving the equations of a cycloid, epicycloid, and hypocycloid
      • ▾Series and sequences
        • •Maclaurin series
        • •Finding the sum of a telescoping series
        • ▸Convergent and divergent geometric series
          • •Lesson
          • •Practice
        • •Alternating harmonic series
        • •Alternating series test
        • •Partial sums
        • •Ratio test
        • ▸Root test
          • •Lesson
          • •Practice
        • •Absolute convergence, conditional convergence, and divergence
        • •Riemann's rearrangement theorem
        • •Taylor series
      • ▸Vectors
        • •Intro to vectors
        • •Length of a vector
        • •Cross product
        • •Intro to the dot product
      • •Competition problems (calculus 2)
     › Calculus 2 › Series and sequences

    Convergent and divergent geometric series

    In this section, you will learn how to determine whether a geometric series converges and diverges. You will also learn how to find the sum of convergent geometric series.

    Conclude by giving your students these challenges:

    • Retiring to Paradise by NRICH
    • What Numbers Can We Make Now? by NRICH
    • Polygon Pictures by NRICH
    • Time of Birth by NRICH
    • Stair Climbing

    Lesson

    Khan Academy videos

    • 2907Worked example: convergent geometric series
      Khan Academy ~ YouTube

    Practice

    This problem is harder, and thus should appear last. It requires the student to split the series in twain, which isn't something they're used to doing.

    For the following convergent series, find the common ratio and the sum of the series.

    \(\displaystyle \sum_{n = 0}^\infty \left(\dfrac{-1}{6}\right)^n\)
    70075Sum of the Series SUM ((-1/6)^n)
    The Math Sorcerer

    For each of the series below, determine whether the series converges. If it does, state the common ratio and the sum of the series. Otherwise, write "undefined."

    \(\dfrac{1}{3} + \dfrac{2}{9} + \dfrac{1}{27} + \dfrac{2}{81} + \dfrac{1}{243} + \dfrac{2}{729} + \ldots\)
    45805Find the Sum of the Infinite Geometric Series Harder Example!
    The Math Sorcerer
    \(\displaystyle \sum_{n = 1}^\infty -2\left(\dfrac{5}{9}\right)^{n - 1}\)
    41851Convergent Geometric Series - Algebra IA 09-0303
    Prof. Redden
    \(\displaystyle \sum_{n = 0}^\infty 3\left(\dfrac{12}{13}\right)^n\)
    41851Convergent Geometric Series - Algebra IA 09-0303
    Prof. Redden