Power Series – Determining Convergence and Interval of Convergence

In a Nut Shell: Infinite series contain a series of constant terms. On the other hand,

a power series contain an infinite number of terms each involving the independent

variable x.

There are two common forms for power series as follows:

∞ ∞ ∞ ∞

∑ a_n xⁿ = ∑ u_n and ∑ a_n ( x – c )ⁿ = ∑ u_n

n=0 n=0 n=0 n=0

The objective is to determine where the power series converges. There are three

possibilities. The power series may converge at only one point, say x = a, it may

converge for all x, or it may converge for some positive value of R such that

the series converges if | x – a | < R and diverges if | x – a | > R.

Strategy: Use the ratio test to evaluate the radius of convergence, R, of a power

series and to find the region of convergence (what values of x that it converges).

Reminder of the Ratio Test for a Series of Constant Terms ∑ a_n

Ratio Test - If P= lim | (a_n+1)/ a_n | exists, then the infinite series

n → ∞

∑ a_n converges absolutely if P < 1, diverges if P > 1 (or P = ∞)

If P = 1 this test fails.

Click here to continue discussion of power series.

Return to Calculus 2 Notes Involving Sequences, Infinite Series, etc

Copyright © 2017 Richard C. Coddington

All rights reserved.