Geometric Calculus

Algebra Level 4

− ( a − b a ) 3 ( a b − c + c c − d + e d − e + g e − f + . . . ) \large - (\frac {a - b}{a})^{3} ( \frac{a}{b-c} + \frac{c}{c-d} + \frac{e}{d-e} + \frac{g}{e-f} + ... )

If a , b , c , d , e . . . a, b, c, d, e ... form a geometric progression with a geometric ratio < 1, find the limit of the above expression as the geometric ratio approaches one half.


The answer is -1.

Efren Medallo by Efren Medallo , 26, Philippines

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1 solution

Efren Medallo
Jun 9, 2015

If we let b b , and all succeeding terms as a r n − 1 ar^{n-1} , for n = 2 , 3 , 4... n = 2, 3, 4 ... , we can simplify the expression as

− ( 1 − r ) 3 ( 1 r − r 2 + r r − r 2 + r 2 r − r 2 + r 3 r − r 2 . . . ) \large -(1-r)^{3} ( \frac {1}{r-r^{2}} + \frac {r}{r-r^{2}} + \frac {r^2}{r-r^{2}} + \frac {r^3}{r-r^{2}} ...)

Now, we can see that the second factor is a geometric series, and simplifying it, we get

− ( 1 − r ) 3 r ( 1 − r ) 2 \large \frac { -(1-r)^{3} } { r(1-r)^{2}} .

r − 1 r \large \frac {r-1 }{r}

1 − 1 r \large 1 - \frac{1}{r}

substituting r = 1 2 r= \frac {1}{2} makes the limit equal to − 1 -1 .

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