Which geometric series converges? StartFraction 1 Over 81 EndFraction + StartFraction 1 Over 27 EndFraction + one-ninth + one-third + ellipsis 1 + one-half + one-fourth + one-eighth + ellipsis Sigma-Summation Underscript n = 1 Overscript infinity EndScripts 7 (negative 4) Superscript n minus 1 Sigma-Summation Underscript n = 1 Overscript infinity EndScripts one-fifth (2) Superscript n minus 1

Question

Mathematics

Savannah Schneider

28 August, 05:31

Which geometric series converges? StartFraction 1 Over 81 EndFraction + StartFraction 1 Over 27 EndFraction + one-ninth + one-third + ellipsis 1 + one-half + one-fourth + one-eighth + ellipsis Sigma-Summation Underscript n = 1 Overscript infinity EndScripts 7 (negative 4) Superscript n minus 1 Sigma-Summation Underscript n = 1 Overscript infinity EndScripts one-fifth (2) Superscript n minus 1

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Mitch · Answer 1

The series converges, and its sum is 1/2.

If r > 1, the series is divergent. If r < 1, the series is convergent. In our sequence, r, the common ratio we multiply by to get the next term, is 7/9; therefore it is convergent.

To find the sum of a convergent series, we use the formula

a / (1-r), where a is the first term and r is the common ratio. We then have

1/9: (1-7/9) = 1/9:2/9 = 1/9*9/2 = 1/2