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Remainders Let Then, the remainder in truncating the power
Chapter 8, Problem 64AE(choose chapter or problem)
Let
\(f(x)=\sum_{k=0}^{\infty} x^{k}=\frac{1}{1-x}\) and \(S_{n}(x)=\sum_{k=0}^{n-1} x^{k}\).
Then, the remainder in truncating the power series after n terms is \(R_{n}=f(x)-S_{n}(x)\), which now depends on x.
a. Show that \(R_{n}(x)=x^{n} /(1-x)\).
b. Graph the remainder function on the interval |x| < 1 for n = 1, 2, 3. Discuss and interpret the graph. Where on the interval is \(\left|R_{n}(x)\right|\) largest? Smallest?
c. For fixed n, minimize \(\left|R_{n}(x)\right|\) with respect to x. Does the result agree with the observations in part (b)?
d. Let N(x) be the number of terms required to reduce \(\left|R_{n}(x)\right|\) to less than \(10^{-6}\). Graph the function N(x) on the interval |x| < 1. Discuss and interpret the graph.
Questions & Answers
QUESTION:
Let
\(f(x)=\sum_{k=0}^{\infty} x^{k}=\frac{1}{1-x}\) and \(S_{n}(x)=\sum_{k=0}^{n-1} x^{k}\).
Then, the remainder in truncating the power series after n terms is \(R_{n}=f(x)-S_{n}(x)\), which now depends on x.
a. Show that \(R_{n}(x)=x^{n} /(1-x)\).
b. Graph the remainder function on the interval |x| < 1 for n = 1, 2, 3. Discuss and interpret the graph. Where on the interval is \(\left|R_{n}(x)\right|\) largest? Smallest?
c. For fixed n, minimize \(\left|R_{n}(x)\right|\) with respect to x. Does the result agree with the observations in part (b)?
d. Let N(x) be the number of terms required to reduce \(\left|R_{n}(x)\right|\) to less than \(10^{-6}\). Graph the function N(x) on the interval |x| < 1. Discuss and interpret the graph.
ANSWER:Solution 64AEStep 1:Given that Let Then, the remainder in truncating the power series after n terms is Rn = f(x) Sn (x), which now depends on x.