9.06 Infinite geometric series
What condition must be satisfied by an infinite geometric series in order for its sum to exist?
For each of the following infinite geometric sequences:
Find the common ratio.
Determine whether the sum is divergent or convergent.
11, \, 22, \, 44, \, 88, \ldots
3, \, -12, \, 48, \, -192, \ldots
-40, \, -20, \, -10, \, -5, \ldots
256, \, 64, \, 16, \, 4, \ldots
For each of the following infinite geometric sequences:
Find the common ratio, r.
Find the limiting sum of the series.
2, \, \dfrac{1}{2}, \, \dfrac{1}{8}, \, \dfrac{1}{32}, \ldots
125, \, 25, \, 5, \, 1, \ldots
16, \, - 8, \, 4, \, - 2, \ldots
28, \, - 7, \, \dfrac{7}{4}, \, - \dfrac{7}{16}, \ldots
Consider the infinite geometric series: 5 + \sqrt{5} + 1 + \ldots
Find the common ratio, r, expressing your answer with a rational denominator.
Find the limiting sum, expressing your answer with a rational denominator.
Consider the infinite geometric series: 6 + 2 + \dfrac{2}{3} + \dfrac{2}{9} + \ldots
Find the number of terms, n, that would be required to give a sum of \dfrac{177\,146}{19\,683}.
Find the infinite sum of the series.
Find the limiting sum of the infinite series \dfrac{1}{5} + \dfrac{3}{5^{2}} + \dfrac{1}{5^{3}} + \dfrac{3}{5^{4}} + \dfrac{1}{5^{5}} + \ldots
For a particular geometric sequence, a = 2 and S_{\infty} = 4.
Find r, the common ratio.
Find the first 3 terms in the sequence.
Consider the recurring decimal 0.2222 \ldots in the form \dfrac{2}{10} + \dfrac{2}{100} + \dfrac{2}{1000} + \dfrac{2}{10\,000} + \ldots Rewrite the recurring decimal as a fraction.
Find the value of the following:
\sum_{i=1}^{\infty} 3 \left(\dfrac{1}{4}\right)^{i - 1}
\sum_{i=1}^{\infty} 5 \left( - \dfrac{1}{4} \right)^{i - 1}
\sum_{i=1}^{\infty} \dfrac{1}{8} \left( - \dfrac{1}{2} \right)^{i - 1}
\sum_{i=1}^{\infty} - \dfrac{1}{2} \left(\dfrac{5}{7}\right)^{i - 1}
\sum_{i=1}^{\infty} \left(0.9\right)^{i}
\sum_{k=1}^{\infty} 9^{ - k }
For each of the following recurring decimals:
Rewrite the decimal as a geometric series.
Express the decimal as a fraction.
0.6666 \ldots
0.066\,66 \ldots
0.151\,515\,15 \ldots
0.021\,212\,1 \ldots
Consider the recurring decimal 0.575\,75 \ldots in the form 0.57 + 0.0057 + \ldots Rewrite the recurring decimal as a fraction.
Consider the sum 0.25 + 0.0025 + 0.000\,025 + \ldots
Express the sum as a decimal.
Hence, express the decimal as a fraction.
Find the value of 0.994 + 0.000\,994 + 0.000\,000\,994 \ldots
Consider a geometric series in which all terms are positive that has first term a and common ratio r. The sum of the infinite series is 768. The sum of the first 4 terms is 765.
Find the first three terms of the sequence.
The limiting sum of the infinite sequence 1, \, 4x, \, 16 x^{2}, \ldots is 5. Find the value of x.
Consider the infinite sequence: 1, x - 7, \left(x - 7\right)^{2}, \left(x - 7\right)^{3}, \ldots
Determine the range of values of x if the infinite series has a limiting sum.