Starting from the Maclauren series for 1/(1-x), find the Maclauren series of ln(1+x^4)
Help would be appreciated, thank you
Help would be appreciated, thank you
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1/(1 - x) = 1 + x + x² + x³ + ... for |x| < 1.
Integrate both sides from 0 to x:
-ln(1 - x) = x + x²/2 + x³/3 + x^4/4 + ...
ln(1 - x) = -(x + x²/2 + x³/3 + x^4/4 + ...)
Replace x by -x^4 on both sides:
ln(1 - (-x^4)) = -((-x^4) + (-x^4)²/2 + (-x^4)³/3 + (-x^4)^4/4 + ...)
ln(1 + x^4) = x^4 - x^8/2 + x^12/3 - x^16/4 +-...
Integrate both sides from 0 to x:
-ln(1 - x) = x + x²/2 + x³/3 + x^4/4 + ...
ln(1 - x) = -(x + x²/2 + x³/3 + x^4/4 + ...)
Replace x by -x^4 on both sides:
ln(1 - (-x^4)) = -((-x^4) + (-x^4)²/2 + (-x^4)³/3 + (-x^4)^4/4 + ...)
ln(1 + x^4) = x^4 - x^8/2 + x^12/3 - x^16/4 +-...