Geometric and Telescoping Series    Lecture 1  Lecture 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) \(\displaystyle{\sum_{n=1}^{\infty}8\left(-\frac{3}{4}\right)^n}\)   \(\displaystyle{\sum_{n=2}^{\infty}8\left(-\frac{3}{4}\right)^n}\)


  3. (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


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


  5. (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


  6. (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


  7. (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


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


  9. \(\displaystyle{\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right)}\)   solution


  10. \(\displaystyle{\sum_{n=3}^{\infty}\left(\frac{1}{n}-\frac{1}{n+2}\right)}\)   solution I   solution II



  11. \(\displaystyle{\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+2}\right)}\)  solution


  12. \(\displaystyle{\sum_{n=1}^{\infty}\left(e^{\frac{1}{n}}-e^{\frac{1}{n+1}}\right)}\)    solution I   solution II


  13. \(\displaystyle{\sum_{n=1}^{\infty}\left(2^{\frac{1}{n}}-2^{\frac{1}{n+1}}\right)}\)  solution


  14. \(\displaystyle{\sum_{n=1}^{\infty} \left(3^{n}-3^{n+1}\right)}\)   solution


  15. \(\displaystyle{\sum_{n=1}^{\infty} \frac{1}{n^2+n}}\)   solution


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


  17. Write the series as a telescoping sum, find a formula for \(S_N\), and find the sum of the series (if it converges).   \(\displaystyle {\sum _{n=2}^\infty \ln \left (1 + \frac {1}{n}\right ) }\)   solution

  18. Find the sum of the geometric series. (a) \(\displaystyle{\sum_{n=2}^{\infty} \frac{5}{2^n}} \)   (b) \(\displaystyle{\sum_{n=1}^{\infty} \frac{6}{(-3)^n}} \)   (c) \(\displaystyle{\sum_{n=1}^{\infty} \frac{4}{2^{-n}}} \)  solution