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Logarithmic Series

Expansion of loge (1 + x); if | x | < 1, then
loge (1 + x) = x87359.png
 
Replacing x by – x in the logarithmic series, we get
loge (1 – x) = – x87365.png
or – loge (1 – x) = x + 87371.png

Some important results from logarithmic series

  1. i. loge (1 + x) + loge (1 – x) = loge (1 – x2)
     
    = –287377.png, (– 1 < x < 1)

    ii. loge (1 + x) – loge (1 – x)
     
    = 287383.png
     
    or loge 87389.png = 87398.png
  2. The series expansion of loge (1 + x) may fail to be valid, if |x| is not less than 1. It can be proved that the logarithmic series is valid for x = 1. Putting x = 1 in the logarithmic series, we get
     
    loge 2 = 1 – 87408.png + … ∞
     
    = 87414.png + … ∞




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