Math Help

Power Series – (continued)

Evaluation of convergence of a power series is typically a three-step process.

Step 1:

Find the radius of convergence of the power series using the ratio test.

Step 2:

Plot the interval of convergence.

Step 3:

Check each end point of the interval for convergence or for divergence.

This step, as before, involves evaluation of an infinite series of constant

terms.

Step 1:

To find the radius of convergence, R, apply the ratio test to the power series.

lim | (u_n+1)/ u_n | = lim | (a_n+1) xⁿ⁺¹/ a_n xⁿ|

n → ∞ n → ∞

which becomes lim | [(a_n+1) / a_n ] x | = P |x| = (1/R) |x|

n → ∞

Note: lim [ (a_n+1) / a_n ] = P which must be less than 1 for convergence.

n → ∞

The radius of convergence is defined to be R = 1/P.

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