Convergence & divergence of telescoping series

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  1. Telescoping Series Overview:
  1. Convergence of Telescoping Series
    Show that the following series are convergent and find its sum:
    1. n=14n2+7n+12\sum_{n=1}^{\infty}\frac{4}{n^2+7n+12}
    2. n=11n2+4n+3\sum_{n=1}^{\infty}\frac{1}{n^2+4n+3}
    3. n=114n21\sum_{n=1}^{\infty}\frac{1}{4n^2-1}
  2. Divergence of Telescoping Series with different pattern
    Show that the series n=1(1)n \sum_{n=1}^{\infty}(-1)^n is a diverging telescoping series.
    Topic Notes
    In this lesson, we will learn about the convergence and divergence of telescoping series. There is no exact formula to see if the infinite series is a telescoping series, but it is very noticeable if you start to see terms cancel out. Most telescopic series problems involve using the partial fraction decomposition before expanding it and seeing terms cancel out, so make sure you know that very well before tackling these questions.
    There is no exact formula for a telescopic series.