How do you test a power series for convergence?

Question

Isaiah Allen · Accepted Answer

Since the convergence of a power series depend on the value of #x#, so the question should be "For which value of #x# does a power series converges?"  For most cases, the ratio test will do the trick. Here is an example. 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)#.

Elijah Abram · Answer

undefined To test a power series for convergence, you can use several methods, including the ratio test, the root test, and the integral test. These tests help determine if a power series converges or diverges based on the properties of its terms.