Geometric and Telescoping Series    Lecture 1Lecture 2

Determine if the series converges or diverges. If it converges, determine its sum.

(1) (a) \(\displaystyle{\sum_{n=0}^{\infty}\left(\frac{2}{3}\right)^n}\)    (b) \(\displaystyle{\sum_{n=2}^{\infty}\left(\frac{1}{2}\right)^n}\)     (c) \(\displaystyle{\sum_{n=0}^{\infty}5\left(-\frac{3}{4}\right)^n}\)     solution


(2) (a) \(\displaystyle{\sum_{n=0}^{\infty}\left(\frac{5}{4}\right)^n}\)   (b) \(\displaystyle{\sum_{n=1}^{\infty}\frac{n}{n+1} }\)     (c) \(\displaystyle \sum_{n=1}^{\infty} 2^{2 n} 3^{1-n}\)   solution


(3) \( \displaystyle{2-\frac{1}{2}+\frac{1}{8}-\frac{1}{32}+\frac{1}{128}-\cdots}\)   solution


(4) (a) \( \displaystyle{\sum_{n=1}^{\infty} \ln \left( \frac{n^2+5}{3n^2+1}\right)}\)   (b) \( \displaystyle{\sum_{n=1}^{\infty} \left(\frac{1}{5^n} + \frac{5}{n}\right)}\)   solution


5. (a) \( \displaystyle \sum_{n=1}^{\infty} \frac{7}{\pi^{n}} \)   (b) \( \displaystyle \sum_{n=1}^{\infty} 8^{n+1} 9^{-n}\)   (c) \( \displaystyle \sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{5^{n}}\)   solution


6. (a) \(\displaystyle{\frac{3}{2}+\frac{3}{4}+\frac{3}{6}+\frac{3}{8}+\cdots}\)   (b) \(\displaystyle{\frac{1}{32}-\frac{1}{16}+\frac{1}{8}-\frac{1}{4}+\cdots}\)  solution


7. \(\displaystyle{ 2-1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\cdots}\)   solution


(8) \(\displaystyle{\sum_{n=3}^{\infty}\left( \frac{1}{n}-\frac{1}{n+2}\right)}\)   solution      (9) \(\displaystyle{\sum_{n=1}^{\infty}\left(e^{\frac{1}{n}}-e^{\frac{1}{n+1}}\right)}\)   solution


(10) \(\displaystyle{\sum_{n=1}^{\infty} \frac{1}{n^2+n}}\)   solution           (11) \(\displaystyle{\sum_{n=2}^{\infty} \frac{n^2+6n+3}{2n^2+n-1}}\)   solution

(11) Find the sum of the geometric series. (a) \(\displaystyle{\sum_{n=2}^{\infty} \frac{5}{2^n}} \)   (b) \(\displaystyle{\sum_{n=1}^{\infty} \frac{6}{(-3)^n}} \)  solution