Using MatLab, Find the interval of convergence for each of the following series.

Question

Using MatLab,

Find the interval of convergence for each of the following series. (Use the usual methods to do so.) Modify the file powseries.m so as to generate two plots: the first plot showing the series and an appropriate partial sum, and the second, in a separate window, showing the absolute difference (error) between the two functions. Select your partial sum so it provides a reasonable approximation to the series throughout most of the interval of convergence. Publish your result. Make sure your graphs have clear titles that explain their content.

Explanation / Answer

To determine the interval of convergence for power series.For power series Summation cnx^n is the interval of convergence consists of interval centered at x=0 including one or both of the endpoints.

Now,for power series summation cn(x-a)^n is the interval of convergence consists of an interval centered at x=a including one or both of endpoints.

a.) inf summation k=1 (-1)^k x^k/k^2      k=1 to infinite

    = -1 x/k^2+ (-1) x^2/k^2+(-1) x^3/k^2+(-1) x^4/k^2    at k=1

    = -x-x^2-x^3-x^4-------

   at k=2

inf summation k=1 (-1)^k x^k/k^2   

= (-1)^2 x^2/2^2

=(-1)^2x^1/4+(-1)^2x^2/4+(-1)^2x^3/4+-----------

=x/4+x^2/4+x^3/4

b.)   inf summation k=0 = x^k/(4+k^2)

      at k=0

      =x^0/(4+k^0)

     = 1/4

       at k=1

      =x^1/(4+k^1)

      =x/(4+k)

       at k = 2

       = x^2/(4+k^2)

       at k=3

       =x^3/(4+k^3)

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