Is there any expansion for log ⁡ ( 1 + x ) when x &gt; 1?Is there any expansion for log ⁡ ( 1 + x ) when x &gt; 1?

Question

Is there any expansion for   log  ⁡  (  1  +  x  ) when   x  &amp;gt;  1?Is there any expansion for   log  ⁡  (  1  +  x  ) when   x  &amp;gt;  1?

bargeolonakc · Accepted Answer

You can expand the function   log  ⁡  (  1  +  x  ) around any point at which it is defined. This means there exists an expansion of   log  ⁡  (  1  +  x  ) around the point   x  =  2, for example, however it will be of the form  log  ⁡  (  1  +  x  )  =      ∑          i      =      1        ∞        a    i    (  x  −  2      )    i    ,and the expansion will be valid for   x  ∈  (  −  1  ,  5  )However, I imagine you want the expansion to of the form  log  ⁡  (  1  +  x  )  =      ∑          i      =      1        ∞        a    i        x    i    ,for which you will have a problem. The problem is that any power series      ∑          i      =      1        ∞        a    i    (  x  −      x    0        )    i  will converge on a interval symmetric around       x    0   (meaning an interval of the type   (      x    0    −  δ  ,      x    0    +  δ  )). This means that if the series expansion for   log  ⁡  (  1  +  x  ) will converge for   x  &amp;gt;  1, it will also converge for some   x  &amp;lt;  −  1 which is impossible.

Deja Bradshaw · Accepted Answer

You are looking for Laurent series for       |    x      |    &amp;gt;  1  ln  ⁡  (  1  +  x  )  =  ln  ⁡  (  x  (  1  +  1      /    x  )  )  =  ln  ⁡  (  x  )  +  ln  ⁡  (  1  +  1      /    x  )  =  ln  ⁡  (  x  )  +      ∑          k      =      1              ∞                  (      −      1              )        k              k        x          −      k      with the condition of convergence       |    x      |    &amp;gt;  1

Is there any expansion for log(1+x) when x>1?

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