Limits as They Relate to Sequences and Series — Quiz

Question 4 • Precalculus

Consider the series 1 eighth plus 1 over 16 plus 1 over 32 plus 1 over 64 plus 1 over 128 plus dot dot dot Which expression defines cap s sub n ?

Answer

limit n to infinity of the fraction with numerator 2 to the n th power minus 1 and denominator 4 times open paren 2 to the n th power close paren

limit n to infinity of the fraction with numerator 1 and denominator 4 times open paren 2 to the n th power close paren

limit n to infinity of the fraction with numerator 1 and denominator 2 to the n th power

limit n to infinity of the fraction with numerator 2 to the n th power and denominator 4 times open paren 2 to the n th power close paren

Explanation

This is a geometric series with first term 1/8 and common ratio 1/2. Its first n terms sum to (1/8)(1 − (1/2)^n)/(1 − 1/2) = (2^n − 1)/(4·2^n). Taking the limit as n approaches infinity gives choice A, which defines the sum of the series.

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