Infinite geometric progression

The sum of infinite number of geometric sequence is infinite geometric sequence. If we have infinite geometric sequence, then $$ \sum_{n=1}^{\infty} a_1q^{n-1} = a_1 + a_1q + a_1q^2 + \cdots,$$ so that \(|q| < 1\), the these sequence is converging and the sum is: $$ S = \frac{a_1}{1-q}. $$

• Example 1 Calculate the value of the expression \(\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3...}}}}}\)
  Solution The mathematical expression can be written as \begin{eqnarray} \sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3...}}}}} &=& 3^{\frac{1}{2}}\cdot 3^{\frac{1}{4}} \cdot 3^{\frac{1}{8}} \cdot 3^{\frac{1}{16}}\cdot 3^{\frac{1}{32}}\cdot ... \\ \sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3...}}}}} &=& 3^{\frac{1}{2} + \frac{1}{4}+ \frac{1}{8} + \frac{1}{16} + \frac{1}{32}+...} \end{eqnarray} By rewritting the expression it can be noticed that the infinite geometric sequence is obtained in exponent. The first member is \(a_1 = \frac{1}{2}\), the quotient is \(q=\frac{1}{2}\) and the sum of the sequence is \begin{eqnarray} S &=& \frac{a_1}{1-q} = \frac{\frac{1}{2}}{1-\frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1 \end{eqnarray} The value of the expression can be calculated as: \begin{eqnarray} 3^{\frac{1}{2} + \frac{1}{4}+ \frac{1}{8} + \frac{1}{16} + \frac{1}{32}+...} &=& 3^S = 3^1 = 3. \end{eqnarray}
• Example 2 Calculate the value \(\sum_{k = 1}^{\infty}4 \cdot (0.5)^k\)
  Solution \begin{eqnarray} \sum_{k = 1}^{\infty}4 \cdot (0.5)^k &=&\\ 4 \cdot \sum_{k=1}^{\infty} (0.5)^k = 4 \frac{\frac{1}{2}}{1-\frac{1}{2}} &=& 4 \end{eqnarray}
• Example 3 Calculate the sum of infinite geometric progression \(1+0.75+0.75^2 +...\)
  Solution The formula for calculating the sum of infinte geometric progression can be written as: \begin{eqnarray} S &=& \frac{a_1}{1-q}. \end{eqnarray} To sole this we have to find parameters \(a_1\) and \(q\). From the given infinite geometric progression it can be seen that \(a_1\) is equal to 1 and \(q\) is equal to 0.75. The formula for calculating the sum is only valid if the infinite geometric progression is converging. The geometric progression is converging if \(|q| < 1\) whcih is true since the value of \(q\) is 0.75. The sum is equal to: \begin{eqnarray} S &=& \frac{a_1}{1-q}\\ S &=& \frac{1}{1-0.75}\\ S &=& \frac{1}{0.25}\\ S &=& \frac{100}{25} = 4 \end{eqnarray}
• Example 4 For the infinite geometric progression calculate the value of first five members. \(5+ 2.5 + 1.25+\cdots\)
  Solution To calculate the value of first five members we need to find all five members. In the problem description only first threre members are given. So, the remaining two must be determine. The missing value in this case is to determine the value of pararmeter \(q\) since the parameter \(a_1\) is the first element of the progression which is equal to 5. \begin{eqnarray} a_1 &=& 5\\ a_1q &=& 2.5\\ q &=& \frac{2.5}{5}\Rightarrow q = 0.5 \end{eqnarray} Since \(q\) is equal to 0.5 the fourth member is \(a_1q^3\) and the fifth member is \(a_1q^4\). The values of forth and fifth member are: \begin{eqnarray} a_1q^3 &=& 5 \cdot 0.5^3 = 0.625,\\ a_1q^4 &=& 5 \cdot 0.5^4 = 0.3125. \end{eqnarray} The sum of first five members is equal to: $$ 1+ 2.5 + 1.25 + 0.625 + 0.3125 = 5.6875$$
• Example 5 If the following infinite geometrci progresion series convergies, calculate the sum. \(5+\frac{15}{4} + \frac{45}{16} + ...\)
  Solution Before we calculate the sum of infinite geometric series we have to investigate if the given progression is converging. The condition for converging is \(|q| < 1.\) From the given progression it can easly be found that \(a_1 = 5\) and \(q=\frac{3}{4}\) Thesting the condtion \begin{eqnarray} q &=& \frac{3}{4}\\ |q| &<& 1 \Rightarrow \left|\frac{3}{4}\right| < 1 \\ 0.75 &<& 1 \end{eqnarray} Since the infinite geometric progression is converging the sum value is equal to: \begin{eqnarray} S &=& \frac{a_1}{1-q}\\ S &=& \frac{5}{1-\frac{3}{4}}\\ S &=& 20 \end{eqnarray}
• Example 6 Check if the following infinite geometric progression convergies. If it converges then calculate the sum. \(1 + \frac{3}{4} + \left(\frac{3}{4}\right)^2 + \left(\frac{3}{4}\right)^3+\cdots\)
  Solution Examining the given inifnite geometric series we can determine the coefficient \(a_1\) and \(q\). The \(a_1\) is equal to 1 while \(q\) is equal to \(\frac{3}{4}\). The value of \(q\) indicates that the infinite geometric series is convergine i.e. the condition \(|q|< 1\) is satisfied. The sum equals to: \begin{eqnarray} S &=& \frac{a_1}{1-q}\\ S &=& \frac{1}{1-\frac{3}{4}}\\ S &=& 4 \end{eqnarray}