title: Bounding a summation by a geometric series
body: <p>I was going through the math/appendix section of the <a href="https://en.wikipedia.org/wiki/Introduction_to_Algorithms" rel="nofollow noreferrer">Introduction to Algorithms</a> book and encountered this proof for bounding by a geometric series:</p>
  <p><a href="https://i.stack.imgur.com/nWk5E.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nWk5E.png" alt="enter image description here" /></a></p>
  <p><strong>And the things that i don't quite understand are:</strong></p>
  <ol>
  <li>If ak+1 / ak &lt;= r and if a2 = 2 and a3 = 3 then how does 0&lt;r&lt;1 hold?</li>
  <li>If the answer to Q1 is that it holds only for a decreasing arithmetic series then how does the summation symbol imply that ?</li>
  <li>What is the difference between bounding each term and bounding the sum of the series?</li>
  <li>How is ak &lt;= a0rk ? Is it a fundamental theorem ?</li>
  </ol>
  <blockquote>
  <p>I know there are many questions here and if you can answer even a single one that would mean a lot because that at least brings me to the right path. Even answer to just references to articles or videos online that can help solve the issue is greatly appreciated.</p>
  </blockquote>
  <blockquote>
  <p>Thank you :D</p>
  </blockquote>