What is the sum of the infinite geometric series 3-1+1/3...?

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What is the sum of the infinite geometric series 3-1+1/3...?

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Answer 1 · 2016-05-21T19:17:40

#3sum_{i=0}^{infty}(-1/3)^i = 9/4#

Explanation:

The sum of sequence #{3, -1,1/3,-1/9,1/27,...} = 3{1,-1/3,1/9,-1/27,...}# or more compactly
#S=3sum_{i=0}^{infty}(-1/3)^i = 3(1/(1+1/3)) = 9/4#
We use here the polynomial identity
#(1-x^{n+1})/(1-x) = 1+x+x^2+x^3+...+x^n#. Here #x = -1/3# and #abs x < 1# so #lim_{n->infty} = 1/(1-x)#

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What is the sum of the infinite geometric series 3-1+1/3...?

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