11.2 Series

Series: the addition of infinitely many terms (i.e., a1+a2+a3+a4+⋯a_1+a_2+a_3+a_4+\cdots )

∑n=1∞an\sum_{n=1}^{\infty}a_{n}






For a series ∑n=1∞an\sum_{n=1}^{\infty}a_{n}, there are 2 associated sequences…

  1. The sequence of terms: a1,a2,a3,…a_1,a_2,a_3,\ldots

  2. The sequence of partial sums: S1+S2+S3+S4+⋯S_1+S_2+S_3+S_4+\cdots

    SN=∑n=0NanS_{N}=\sum_{n=0}^{N}a_{n}







Geometric Series

Given a series that follows ∑n=0∞arn\sum_{n=0}^{\infty}ar^{n} with ∣r∣<1\left|r\right|<1 , that series converges to a1−r\frac{a}{1-r}

(ex) ∑n=0∞(12)(14)n\sum_{n=0}^{\infty}\left(\frac12\right)\left(\frac14\right)^{n}                                                      (ex) ∑n=1∞(2)(14)n\sum_{n=1}^{\infty}\left(2\right)\left(\frac14\right)^{n}







Test for Divergence

If lim⁡n→∞an≠0\lim_{n\to\infty}a_{n}\ne0 of a series ∑n=1∞an\sum_{n=1}^{\infty}a_{n} , that series diverges

(ex) ∑n=1∞2+n1−2n\sum_{n=1}^{\infty}\frac{2+n}{1-2n}                                                              (ex) ∑n=1∞2+n1−2n2\sum_{n=1}^{\infty}\frac{2+n}{1-2n^2}







Telescoping Series

Sums in which pairs of consecutive terms partly cancel each other

(ex) ∑n=2∞2n2−1\sum_{n=2}^{\infty}\frac{2}{n^2-1}