What is a power series representation for f ( x ) = ln ( 1 + x ) and what is its radius of convergence?

Question

What is a power series representation for             f              (        x        )              =          ln              (        1        +        x        )             and what is its radius of convergence?

leoanardig7k · Accepted Answer

Start from the sum of the geometric series:                    ∑                  n          =          0                ∞                    q      n        =          1              1        -        q            converging for             |      q      |        &amp;lt;    1  .Let       x    =    -    q   to have:                    ∑                  n          =          0                ∞                            (        -        1        )            n              x      n        =          1              1        +        x            Inside the interval of convergence       x    ∈          (      -      1      ,      1      )       we can integrate the series term by term:                    ∫        0        x                            d        t                    1        +        t              =                  ∑                  n          =          0                ∞                            ∫        0        x                            (        -        1        )            n              t      n              d      t      and obtain a series with the same radius of convergence       R    =    1  :            ln              (        1        +        x        )              =                  ∑                  n          =          0                ∞                            (        -        1        )            n                      x                  n          +          1                            n        +        1

What is a power series representation for f(x)=ln(1+x) and what is its radius of convergence?

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