Barrons AP Calculus - David Bock

see that the error is always less than that next term. Since |Rn | < an + 1, the error bound for an
alternating series is the first term omitted or dropped.

EXAMPLE 31

The series passes the Alternating Series Test; hence its sum differs from the sum

by less than which is the error bound.

EXAMPLE 32

How many terms must be summed to approximate to three decimal places the value of

SOLUTION: Since the series converges by the Alternating Series Test;

therefore after summing a number of terms the remainder (error) will be less than the first omitted

term.

We seek n such that Thus n must satisfy (n + 1)^2 > 1000, or n > 30.623. Therefore

31 terms are needed for the desired accuracy.

C. POWER SERIES

C1. Definitions; Convergence.

An expression of the form

where the a’s are constants, is called a power series in x; and

is called a power series in (x − a).
If in (1) or (2) x is replaced by a specific real number, then the power series becomes a series of
constants that either converges or diverges. Note that series (1) converges if x = 0 and series (2)
converges if x = a.

RADIUS AND INTERVAL OF CONVERGENCE
If power series (1) converges when |x| < r and diverges when |x| > r, then r is called the radius of
convergence. Similarly, r is the radius of convergence of power series (2) if (2) converges when |x −
a| < r and diverges when |x − a| > r.