Notes

 

Questions

Find the infinite geometric series

\(\textbf{1)}\) \(a_1=8 \,\, r=.75 \)

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The answer is \(32\)

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\(\,\,\,\,\,\,S_\infty=\frac{a_1}{1-r},\,\,\,|r|\lt 1\)
\(\,\,\,\,\,\,|.75|\lt 1\)
\(\,\,\,\,\,\,S_\infty=\frac{8}{1-.75}\)
\(\,\,\,\,\,\,S_\infty=\frac{8}{.25}\)
\(\,\,\,\,\,\,S_\infty=32\)

 

\(\textbf{2)}\) \(a_1=.75 \,\, r=8 \)

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The answer is Divergent

 

\(\textbf{3)}\) \(\displaystyle\frac{1}{2}+\displaystyle\frac{1}{4}+\displaystyle\frac{1}{8}+\displaystyle\frac{1}{16}+ \cdots\)

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The answer is \(1\)

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\(\,\,\,\,\,\,S_\infty=\frac{a_1}{1-r},\,\,\,|r|\lt 1\)
\(\,\,\,\,\,\,a_1=\frac{1}{2},\,\,\,r=\frac{1/4}{1/2}=\frac{1}{2} \)
\(\,\,\,\,\,\,|\frac{1}{2}|\lt 1\)
\(\,\,\,\,\,\,S_\infty=\frac{\frac{1}{2}}{1-\frac{1}{2}}\)
\(\,\,\,\,\,\,S_\infty=\frac{\frac{1}{2}}{\frac{1}{2}}\)
\(\,\,\,\,\,\,S_\infty=1\)

 

\(\textbf{4)}\) \(\displaystyle\frac{1}{2}-\displaystyle\frac{1}{4}+\displaystyle\frac{1}{8}-\displaystyle\frac{1}{16}+ \cdots\)

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The answer is \(\displaystyle\frac{1}{3}\)

 

\(\textbf{5)}\) \(\displaystyle\frac{1}{20}-\displaystyle\frac{1}{10}+\displaystyle\frac{1}{5}-\displaystyle\frac{2}{5}+ \cdots\)

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The answer is Divergent

 

\(\textbf{6)}\) \( \displaystyle \sum_{i=1}^{\infty} 4\left( \frac{1}{3} \right)^{i-1} \)

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The answer is \( 6 \)

 

See Related Pages

\(\bullet\text{ Arithmetic Sequences}\)
\(\,\,\,\,\,\,\,a_n=a_1 + d(n-1)\)

\(\bullet\text{ Geometric Sequences}\)
\(\,\,\,\,\,\,\,a_n=a_1 \cdot r^{(n-1)}…\)

\(\bullet\text{ Arithmetic Series}\)
\(\,\,\,\,\,\,\,s_n=\frac{n}{2}(a_1+a_n)…\)

\(\bullet\text{ Geometric Series}\)
\(\,\,\,\,\,\,\,s_n=a_1 \frac{1-r^n}{1-r}…\)

\(\bullet\text{ Summation Notation}\)
\(\,\,\,\,\,\,\, \displaystyle \sum_{i=4}^{9} 3i-5 …\)

\(\bullet\text{ Recursive Sequences}\)
\(\,\,\,\,\,\,\, a_{1}=2, \, a_{n+1}=a_{n}+3…\)