Approximating a Sum In Exercises 31 and 32, (a) use a graphing utility to graph several partial sums of the series. b) find the sum of the series and its radius of convergence,(c) use a graphing utility and 50 terms of the series to approximate the sum when x = 0.5 . and (d) determine what the approximation represents and how good the approximation is. ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) !
Solution Summary: The author illustrates the graph obtained by the use of a graphing utility.
Approximating a Sum In Exercises 31 and 32, (a) use a graphing utility to graph several partial sums of the series. b) find the sum of the series and its radius of convergence,(c) use a graphing utility and 50 terms of the series to approximate the sum when
x
=
0.5
. and (d) determine what the approximation represents and how good the approximation is.
Explore the Alternating Series Remainder. (a) Use a graphing utility to find the indicated partial sum Sn and complete the table. (b) Use a graphing utility to graph the first 10 terms of the sequence of partial sums and a horizontal line representing the sum. (c) What pattern exists between the plot of the successive points in part (b) relative to the horizontal line representing the sum of the series? Do the distances between the successive points and the horizontal line increase or decrease? (d) Discuss the relationship between the answers in part (c) and the Alternating Series Remainder
Proving the Alternating Series Test (Theorem 2.7.7) amountsto showing that the sequence of partial sums
sn = a1 − a2 + a3 −· · ·±an
converges. (The opening example in Section 2.1 includes a typical illustration of (sn).) Different characterizations of completeness lead to different proofs.
(a) Prove the Alternating Series Test by showing that (sn) is a Cauchysequence.
(b) Supply another proof for this result using the Nested Interval Property(Theorem 1.4.1).
(c) Consider the subsequences (s2n) and (s2n+1), and show how the Monotone Convergence Theorem leads to a third proof for the Alternating Series Test.
Proof In Exercises 17–20, prove that the Maclaurin series for the function converges to the function for all x.
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