What is interval of convergence for a Power Series?
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"Suppose that I don't have a formula for #g(x)# but I know that #g(1)
= 3# and #g'(x) = sqrt(x^2+15)# for all x. How do I use a linear approximation to estimate #g(0.9)# and #g(1.1)#?"
The interval of convergence of a power series is the set of all x-values for which the power series converges.
Let us find the interval of convergence of #sum_{n=0}^infty{x^n}/n#.
By Ratio Test,
#lim_{n to infty}|{a_{n+1}}/{a_n}|
=lim_{n to infty}|x^{n+1}/{n+1}cdotn/x^n|
=|x|lim_{n to infty}n/{n+1}#
#=|x|cdot 1=|x|<1 Rightarrow -1 < x < 1#,
which means that the power series converges at least on #(-1,1)#.
Now, we need to check its convergence at the endpoints: #x=-1# and #x=1#.
If #x=-1#, the power series becomes the alternating harmonic series
#sum_{n=0}^infty(-1)^n/n#,
which is convergent. So, #x=1# should be included.
If #x=1#, the power series becomes the harmonic series
#sum_{n=0}^infty1/n#,
which is divergent. So, #x=1# should be excluded.
Hence, the interval of convergence is #[-1,1)#.
