Infinite geometric series: the sum to infinity Problem set A One question at a time

Find the sum to infinity of each series.

1. (a) \(\frac34+\frac{3}{16}+\frac{3}{64}+\dots\)

   • \(u_1=\frac{3}{4},\ r=\frac{1}{4}\)
   • \(|r|<1\), so \(S_\infty=\frac{\frac{3}{4}}{1-\frac{1}{4}}\)
   • \(S_\infty=1\)

2. (b) \(18+12+8+\dots\)

   • \(u_1=18,\ r=\frac{2}{3}\)
   • \(|r|<1\), so \(S_\infty=\frac{18}{1-\frac{2}{3}}\)
   • \(S_\infty=54\)

3. (c) \(10-5+2.5-1.25+\dots\)

   • \(u_1=10,\ r=-\frac{1}{2}\)
   • \(|r|<1\), so \(S_\infty=\frac{10}{1-(-\frac{1}{2})}\)
   • \(S_\infty=\frac{20}{3}\)

Does the series \(4+5+6.25+\dots\) have a sum to infinity? Give a reason.

• \(r=\frac{5}{4}=1.25\)
• \(\frac{\frac{25}{4}}{5}=1.25\), so the series is geometric
• \(|r|>1\): the terms grow, so there is no sum to infinity

Write \(0.\overline{27}\) as a fraction in lowest terms.

• \(0.\overline{27}=27/100+27/10\,000+\dots\), so \(u_1=\frac{27}{100},\ r=\frac{1}{100}\)
• \(S_\infty=\frac{27/100}{99/100}=\frac{27}{99}\)
• Lowest terms: \(\frac{3}{11}\)

Write each recurring decimal as a fraction in lowest terms.

1. (a) \(0.\overline{8}\)

   • \(u_1=\frac{8}{10},\ r=\frac1{10}\)
   • \(S_\infty=\frac{8/10}{9/10}=\frac{8}{9}\)

2. (b) \(0.\overline{135}\)

   • \(u_1=\frac{135}{1000},\ r=\frac{1}{1000}\)
   • \(S_\infty=\frac{135}{999}\)
   • Lowest terms: \(\frac{5}{37}\)

A geometric series has second term 6 and sum to infinity 25. Find the two possible values of the common ratio.

• \(u_1=\frac{6}{r}\), so \(\frac{6/r}{1-r}=25\)
• \(6=25r(1-r)\), so \(25r^2-25r+6=0\)
• From the calculator: \(r=0.4\) or \(r=0.6\)
• Both have \(|r|<1\), so both are allowed

A trust pays a scholarship of 500 euros in the first year. Each later year the payment is 80% of the payment the year before, and this goes on for ever. Find the total the trust pays out.

• A geometric series: \(u_1=500,\ r=0.8\)
• \(|r|<1\), so \(S_\infty=\frac{500}{1-0.8}\)
• \(S_\infty=2500\)

A pendulum swings through an arc of 40 cm on its first swing. Each swing is 90% as long as the swing before. Find the total distance the pendulum travels before it stops.

• A geometric series: \(u_1=40,\ r=0.9\)
• \(S_\infty=\frac{40}{1-0.9}\)
• \(S_\infty=400\) cm

A geometric series has second term \(-6\) and sum to infinity 8. Find the common ratio and the first term.

• \(u_1=\frac{-6}{r}\), so \(-6=8r(1-r)\) and \(8r^2-8r-6=0\)
• From the calculator: \(r=-\frac{1}{2}\) or \(r=\frac{3}{2}\)
• \(|\frac{3}{2}|>1\), so reject it. \(r=-\frac{1}{2}\)
• \(u_1=\frac{-6}{-\frac{1}{2}}=12\)

Write \(0.3\overline{18}\) as a fraction in lowest terms. Only the digits shown under the bar repeat.

• \(0.3\overline{18}=0.3+0.018+0.00018+\dots\)
• The series after 0.3 has \(u_1=0.018=\frac{18}{1000},\ r=\frac1{100}\)
• \(S_\infty=\frac{18/1000}{99/100}=\frac{18}{990}=\frac1{55}\)
• \(\frac3{10}+\frac1{55}=\frac{33}{110}+\frac{2}{110}=\frac{35}{110}=\frac7{22}\)