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Chapter 10: Problem 27
Find the sum of the convergent series. $$ \sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}-\frac{1}{3^{n}}\right) $$
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The repeating decimal is expressed as a geometric series. Find the sum of the geometric series and write the decimal as the ratio of two integers. $$ 0 . \overline{9}=0.9+0.09+0.009+0.0009+\cdots $$
Use a symbolic algebra utility to find the sum of the convergent series. $$ \sum_{n=0}^{\infty}\left(-\frac{1}{2}\right)^{n}=1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots $$
Find the sum of the convergent series. $$ \sum_{n=0}^{\infty} 4\left(\frac{1}{4}\right)^{n}=4+1+\frac{1}{4}+\frac{1}{16}+\cdots $$
Determine whether the sequence is arithmetic or geometric, and write the \(n\) th term of the sequence. $$ 20,10,5, \frac{5}{2}, \dots $$
Verify that the infinite series diverges. $$ \sum_{n=0}^{\infty}\left(\frac{4}{3}\right)^{n}=1+\frac{4}{3}+\frac{16}{9}+\frac{64}{27}+\cdots $$
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