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(Solved): Please provide all steps used to arrive at the answer, and complete a, b and c. Please ensure steps ...


(a) Find a series in powers of \( x \) for the function \( f(x)=x \ln \left(1+x^{3}\right) \). (b) Use the result of (a) to f

Please provide all steps used to arrive at the answer, and complete a, b and c. Please ensure steps are easy to read.

(a) Find a series in powers of for the function . (b) Use the result of (a) to find an alternating series that converges to . (c) Use the alternating series error estimate to find an approximation to the integral accurate to within .


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(a): To find a series representation for the function   , we can use the power series expansion of   .

The power series expansion of    is given by:

  

Now, let's substitute    in place of    in the expansion:

  

Finally, multiply the series by    to obtain the series representation for   

  
Expanding and simplifying:

  

Therefore, the series representation for the function    is:

  
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