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Question
Write the first 4 terms of the logarithmic series
log(1 + 4x) Find the intervals on which the expansions are valid.
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Solution
lo(1 + x) = `x - x^2/2 + x^3/3 - x^4/4 ...`
log(1 – x) = `x - x^2/2 - x^3/3 ...`
log(1 + 4x) = `4x - (4x)^2/2 + (4x)^3/3 - (4x)^4/4 ...`
Hence |4x| < 1
⇒ |x| < `1/4`
= `4x - (16x^2)/2 + (64x^3)/3 - (256x^4)/4 ...`
= `4x - 8x^2 + 64/3 x^3 - 64x^4 ...`
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