{"id":1671,"date":"2025-07-25T19:12:57","date_gmt":"2025-07-25T19:12:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1671"},"modified":"2026-03-26T16:07:15","modified_gmt":"2026-03-26T16:07:15","slug":"series-and-their-notations-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/series-and-their-notations-learn-it-4\/","title":{"raw":"Series and Their Notations: Learn It 4","rendered":"Series and Their Notations: Learn It 4"},"content":{"raw":"

Infinite Geometric Series<\/h2>\r\nThus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first\u00a0n<\/em> terms. An\u00a0infinite series<\/strong>\u00a0is the sum of the terms of an infinite sequence. An example of an infinite series is [latex]2+4+6+8+\\dots[\/latex].\r\n\r\nThis series can also be written in summation notation as [latex] \\sum\\limits _{k=1}^{\\infty} 2k[\/latex],\u00a0where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series\u00a0diverges<\/strong>.\r\n\r\nIf the terms of an\u00a0infinite geometric series<\/span> approach [latex]0[\/latex], the sum of an infinite geometric series can be defined. As\u00a0[latex]n[\/latex] gets large, the values of of [latex]r^n[\/latex] get very small and approach [latex]0[\/latex].\u00a0Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to [latex]0[\/latex], the sum of the terms approaches a finite value. The terms of any infinite geometric series with [latex]-1 < r < 1[\/latex]\u00a0approach [latex]0[\/latex]; the sum of a geometric series is defined when\u00a0[latex]-1 < r < 1[\/latex].\r\n\r\n
\r\n
\r\n

determining whether the sum of an infinite geometric series is defined<\/h3>\r\nIf the terms<\/strong> of an\u00a0infinite geometric series\u00a0approach [latex]0[\/latex], the sum of an infinite geometric series can be defined.\r\n\r\n \r\n\r\nThe sum<\/strong> of an infinite series is defined if the series is geometric and\u00a0[latex]-1 < r < 1[\/latex].\r\n\r\n<\/div>\r\n<\/section>
The terms in this series approach [latex]0[\/latex]:\r\n

[latex]1+0.2+0.04+0.008+0.0016+\\dots[\/latex]<\/p>\r\nThe common ratio is [latex]r=0.2[\/latex].\r\n\r\n<\/section>

How To: Given the first several terms of an infinite series, determine if the sum of the series exists.<\/strong>\r\n
    \r\n \t
  1. Find the ratio of the second term to the first term.<\/li>\r\n \t
  2. Find the ratio of the third term to the second term.<\/li>\r\n \t
  3. Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.<\/li>\r\n \t
  4. If a common ratio,\u00a0[latex]r[\/latex], was found in step 3, check to see if [latex]-1 < r < 1[\/latex]. If so, the sum is defined. If not, the sum is not defined.<\/li>\r\n<\/ol>\r\n<\/section>
    Determine whether the sum of each infinite series is defined.\r\n
      \r\n \t
    1. [latex]12+8+4+\\dots[\/latex]<\/li>\r\n \t
    2. [latex]\\dfrac{3}{4}+\\dfrac{1}{2}+\\dfrac{1}{3}+\\dots[\/latex]<\/li>\r\n \t
    3. [latex]\\sum\\limits _{k=1}^{\\infty}{27}\\cdot\\left(\\dfrac{1}{3}\\right)^k[\/latex]<\/li>\r\n \t
    4. [latex]\\sum\\limits _{k=1}^{\\infty}{5k}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"250515\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"250515\"]\r\n
        \r\n \t
      1. The ratio of the second term to the first is [latex]\\frac{2}{3}[\/latex], which is not the same as the ratio of the third term to the second, [latex]\\frac{1}{2}[\/latex].\u00a0The series is not geometric.<\/li>\r\n \t
      2. The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of [latex]\\frac{2}{3}[\/latex]. The sum of the infinite series is defined.<\/li>\r\n \t
      3. The given formula is exponential with a base of [latex]\\frac{1}{3}[\/latex]; the series is geometric with a common ratio of\u00a0[latex]\\frac{1}{3}[\/latex]. The sum of the infinite series is defined.<\/li>\r\n \t
      4. The given formula is not exponential. The series is arithmetic, not geometric and so cannot yield a finite sum.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n

        Finding Sums of Infinite Series<\/h2>\r\nWhen the sum of an infinite geometric series exists, we can calculate the sum.\r\n

        The formula for the sum of an infinite series is related to the formula for the sum of the first [latex]n[\/latex]\u00a0terms of a geometric series.<\/p>\r\n

        [latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex]<\/p>\r\nAs\u00a0[latex]n[\/latex] gets large, [latex]r^n[\/latex] gets very small. We say that as\u00a0[latex]n[\/latex] increases without bound,\u00a0[latex]r^n[\/latex] approaches [latex]0[\/latex]. As\u00a0[latex]r^n[\/latex] approaches [latex]0[\/latex],\u00a0[latex]1-r^n[\/latex] approaches [latex]1[\/latex]. When this happens the numerator approaches [latex]a_1[\/latex]. This gives us the formula for the sum of an infinite geometric series.\r\n\r\n

        \r\n

        formula for the sum of an infinite geometric series<\/h3>\r\nThe formula for the sum of an infinite geometric series with [latex]-1 < r < 1[\/latex] is\r\n

        [latex]S=\\dfrac{{a}_{1}}{1-r}[\/latex]<\/p>\r\n\r\n<\/section>

        We will examine an infinite series with [latex]r=\\frac{1}{2}[\/latex]. What happens to [latex]r^n[\/latex] as\u00a0[latex]n[\/latex] increases?\r\n

        [latex]\\begin{align} &{\\left(\\frac{1}{2}\\right)}^{2} = \\frac{1}{4} \\\\&{\\left(\\frac{1}{2}\\right)}^{3} = \\frac{1}{8} \\\\&{\\left(\\frac{1}{2}\\right)}^{4} = \\frac{1}{16} \\end{align}[\/latex]<\/p>\r\nThe value of [latex]r^n[\/latex] decreases rapidly. What happens for greater values of\u00a0[latex]n[\/latex]?\r\n

        [latex]\\begin{align} &{\\left(\\frac{1}{2}\\right)}^{10} = \\frac{1}{1\\text{,}024} \\\\&{\\left(\\frac{1}{2}\\right)}^{20} = \\frac{1}{1\\text{,}048\\text{,}576} \\\\&{\\left(\\frac{1}{2}\\right)}^{30} = \\frac{1}{1\\text{,}073\\text{,}741\\text{,}824} \\end{align}[\/latex]<\/p>\r\n\r\n<\/section>

        How To: Given an infinite geometric series, find its sum.<\/strong>\r\n
          \r\n \t
        1. Identify [latex]a_1[\/latex] and\u00a0[latex]r[\/latex].<\/li>\r\n \t
        2. Confirm that [latex]-1 < r < 1[\/latex].<\/li>\r\n \t
        3. Substitute values for\u00a0[latex]a_1[\/latex] and\u00a0r<\/em> into the formula,\u00a0[latex]S=\\dfrac{{a}_{1}}{1-r}[\/latex].<\/li>\r\n \t
        4. Simplify to find\u00a0[latex]S[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
          Find the sum, if it exists, for the following:\r\n
            \r\n \t
          1. [latex]10+9+8+7+\\dots[\/latex]<\/li>\r\n \t
          2. [latex]248.6+99.44+39.776+\\dots[\/latex]<\/span><\/li>\r\n \t
          3. [latex]\\sum\\limits _{k=1}^{\\infty}4\\text{,}374\\cdot\\left(-\\dfrac{1}{3}\\right)^{k-1}[\/latex]<\/li>\r\n \t
          4. [latex]\\sum\\limits _{k=1}^{\\infty}\\dfrac{1}{9}\\cdot\\left(\\dfrac{4}{3}\\right)^{k}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"18513\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"18513\"]\r\n
              \r\n \t
            1. There is not a constant ratio; the series is not geometric.<\/li>\r\n \t
            2. There is a constant ratio; the series is geometric. [latex]a_1=248.6[\/latex] and [latex]r=\\dfrac{99.44}{248.6}=0.4[\/latex], so the sum exists. Substitute\u00a0[latex]a_1=248.6[\/latex] and [latex]r=0.4[\/latex] into the formula and simplify to find the sum.\r\n
              [latex]\\begin{align} \\\\ &S=\\frac{a_1}{1-r} \\\\[1.5mm] &S=\\frac{248.6}{1-0.4}=\\frac{1243}{3} \\\\ \\text{ }\\end{align}[\/latex]<\/center><\/li>\r\n \t
            3. The formula is exponential, so the series is geometric with [latex]r=-\\frac{1}{3}[\/latex]. Find [latex]a_1[\/latex] by substituting [latex]k=1[\/latex] into the given explicit formula.\r\n
              [latex]\\begin{align} \\\\ a_1=4\\text{,}374\\cdot\\left(-\\frac{1}{3}\\right)^{1-1}=4\\text{,}374 \\\\ \\text{ }\\end{align}[\/latex]<\/center>Substitute [latex]4\\text{,}374[\/latex] and [latex]r=-\\frac{1}{3}[\/latex] into the formula, and simplify to find the sum.\r\n
              [latex]\\begin{align}\\\\&S=\\frac{a_1}{1-r} \\\\[1.5mm] &S=\\frac{4\\text{,}374}{1-\\left(-\\frac{1}{3}\\right)}=3\\text{,}280.5 \\\\ \\text{ }\\end{align}[\/latex]<\/center><\/li>\r\n \t
            4. The formula is exponential, so the series is geometric, but [latex]r>1[\/latex].\u00a0The sum does not exist.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
              Find the equivalent fraction for the repeating decimal [latex]0.\\overline{3}[\/latex].[reveal-answer q=\"624358\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624358\"]We notice the repeating decimal [latex]0.\\overline{3}=0.333\\dots[\/latex].\u00a0so we can rewrite the repeating decimal as a sum of terms.\r\n

              [latex]0.\\overline{3}=0.3+0.03+0.003+\\dots[\/latex]<\/p>\r\nLooking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to [latex]0.1[\/latex] in the second term, and the second term multiplied to [latex]0.1[\/latex] in the third term.\r\n

              [latex]\\begin{align}0.\\overline{3}&=0.3+0.3\\cdot(0.1)+0.3\\cdot(0.01)+0.3\\cdot(0.001)+\\dots \\\\ &=0.3+0.3\\cdot(0.1)+0.3\\cdot(0.1)^2+0.3\\cdot(0.1)^3+\\dots\\end{align}[\/latex]<\/p>\r\nNotice the pattern; we multiply each consecutive term by a common ratio of [latex]0.1 [\/latex] starting with the first term of [latex]0.3[\/latex]. So, substituting into our formula for an infinite geometric sum, we have\r\n

              [latex]S=\\dfrac{a_1}{1-r} =\\dfrac{0.3}{1-0.1} =\\dfrac{0.3}{0.9} =\\dfrac{1}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

              [ohm_question hide_question_numbers=1]321983[\/ohm_question]<\/section>
              [ohm_question hide_question_numbers=1]321984[\/ohm_question]<\/section>","rendered":"

              Infinite Geometric Series<\/h2>\n

              Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first\u00a0n<\/em> terms. An\u00a0infinite series<\/strong>\u00a0is the sum of the terms of an infinite sequence. An example of an infinite series is [latex]2+4+6+8+\\dots[\/latex].<\/p>\n

              This series can also be written in summation notation as [latex]\\sum\\limits _{k=1}^{\\infty} 2k[\/latex],\u00a0where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series\u00a0diverges<\/strong>.<\/p>\n

              If the terms of an\u00a0infinite geometric series<\/span> approach [latex]0[\/latex], the sum of an infinite geometric series can be defined. As\u00a0[latex]n[\/latex] gets large, the values of of [latex]r^n[\/latex] get very small and approach [latex]0[\/latex].\u00a0Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to [latex]0[\/latex], the sum of the terms approaches a finite value. The terms of any infinite geometric series with [latex]-1 < r < 1[\/latex]\u00a0approach [latex]0[\/latex]; the sum of a geometric series is defined when\u00a0[latex]-1 < r < 1[\/latex].\n\n\n\n

              \n
              \n

              determining whether the sum of an infinite geometric series is defined<\/h3>\n

              If the terms<\/strong> of an\u00a0infinite geometric series\u00a0approach [latex]0[\/latex], the sum of an infinite geometric series can be defined.<\/p>\n

               <\/p>\n

              The sum<\/strong> of an infinite series is defined if the series is geometric and\u00a0[latex]-1 < r < 1[\/latex].\n\n<\/div>\n<\/section>\n

              The terms in this series approach [latex]0[\/latex]:<\/p>\n

              [latex]1+0.2+0.04+0.008+0.0016+\\dots[\/latex]<\/p>\n

              The common ratio is [latex]r=0.2[\/latex].<\/p>\n<\/section>\n

              How To: Given the first several terms of an infinite series, determine if the sum of the series exists.<\/strong><\/p>\n
                \n
              1. Find the ratio of the second term to the first term.<\/li>\n
              2. Find the ratio of the third term to the second term.<\/li>\n
              3. Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.<\/li>\n
              4. If a common ratio,\u00a0[latex]r[\/latex], was found in step 3, check to see if [latex]-1 < r < 1[\/latex]. If so, the sum is defined. If not, the sum is not defined.<\/li>\n<\/ol>\n<\/section>\n
                Determine whether the sum of each infinite series is defined.<\/p>\n
                  \n
                1. [latex]12+8+4+\\dots[\/latex]<\/li>\n
                2. [latex]\\dfrac{3}{4}+\\dfrac{1}{2}+\\dfrac{1}{3}+\\dots[\/latex]<\/li>\n
                3. [latex]\\sum\\limits _{k=1}^{\\infty}{27}\\cdot\\left(\\dfrac{1}{3}\\right)^k[\/latex]<\/li>\n
                4. [latex]\\sum\\limits _{k=1}^{\\infty}{5k}[\/latex]<\/li>\n<\/ol>\n