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Binomial Theorem for Any Index

Let n be a rational number and x be a real number such that |x| < 1, then
 
67050.png
67044.png xr + ... ∞
 
Notes:
  1. The condition |x| < 1 is unnecessary, if nis a whole number while the same condition is essential if n is a rational number other than a whole number.
  2. Note that there are infinite number of term in the expansion of (1 + x)n, whenn is a negative integer or a fraction.
  3. In the above expansion the first term is unity, If the first term is not unity and the index of the binomial is either a negative integer or a fraction, then we expand as follows:
     
    68058.png
     
    68051.png
     
    68045.png
     
    The expansion is valid when 68039.png < 1 or equivalently |x| < |a|.
  4. Expansion of (x + a)n for any rational index
     
    Case I When x > a, i.e., 68032.png
     
    In this case,
     
    68026.png
     
    68020.png
     
    Case II When x < a, i.e., 68014.png
     
    In this case,
     
    68459.png
     
    68447.png
  5. If n is a positive integer the above expansion contains (n + 1) terms and coincides with
     
    (1 + x)n = nC0 + nC1x + nC2 x2 + ... + nCn xn, because nC0 = 1, nC1 = n68358.png
     
    The general term in the expansion of (1 +x)n is given by 68352.png
  6. Let n is positive integer then by replacing n by – n in the expansion for (1 + x)n, we get 68346.png
    = 1 – nC1x + n+1C2x2 – n+2C3x3 + ... +n+r–1Cr(–x)r + ...
     
    Now replacing x by –x and n by –n in the expression of (1 + x)n, we get
    68339.png
    = 1 + nC1x + n+1C2x2 + n+2C3x3 + ... + n+r–1Crxr + ...




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