summed to approximate to three decimal places the value of
. . . ?SOLUTION: Since and , the series converges by the
Alternating Series Test; therefore after summing a number of terms the
remainder (alternating series error bound) 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.
Interval of convergenceThe set of all values of x for which a power series converges is called its
interval of convergence. To find the interval of convergence, first determine the
radius of convergence by applying the Ratio Test to the series of absolute values.
Then check each endpoint to determine whether the series converges or diverges
there.