anonymous
  • anonymous
Determine whether each infinite geometric series diverges or converges. Find the sum if the series converges. 1/2 + 1/16 + 1/128 + ...
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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anonymous
  • anonymous
\[\frac{1}{2}+\frac{1}{16}+\frac{1}{128}+\cdots=\left(\frac{1}{2}\right)^1+\left(\frac{1}{2}\right)^4+\left(\frac{1}{2}\right)^7+\cdots=\sum_{n=0}^\infty\left(\frac{1}{2}\right)^{1+3n}\]
campbell_st
  • campbell_st
find the common ratio of the series... if -1 < r < 1 then the series converges if it does converge the limiting sum is is \[S_{\infty} = \frac{a}{1 - r}\]
anonymous
  • anonymous
What's the common ratio?

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campbell_st
  • campbell_st
wow its interesting you are doing a geometric series question and don't know what the common ratio is. you can find it by comparing the ratio of terms \[\frac{a_{2}}{a_{1}} = \frac{a_{3}}{a_{2}} = ... \] \[\frac{\frac{1}{16}}{\frac{1}{2}} = \frac{\frac{1}{128}}{\frac{1}{16}}\] the alternate method is to use \[a_{n} = a_{1} r^{n -1}\] you know the 1st term is 1/2 and the 2nd term is 1/16 then \[\frac{1}{16} = \frac{1}{2} \times r^{(2-1)}\] solve for r

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