Question

{Use of Tech} Remainders Let

f(x) = ∑ₖ₌₀∞ xᵏ = 1/(1−x) and Sₙ(x) = ∑ₖ₌₀ⁿ⁻¹ xᵏ

The remainder in truncating the power series after n terms is Rₙ = f(x) − Sₙ(x), which depends on x.

a. Show that Rₙ(x) = xⁿ /(1−x).
b. Graph the remainder function on the interval |x| < 1, for n=1, 2, and 3 . Discuss and interpret the graph. Where on the interval is |Rₙ(x)| largest? Smallest?
c. For fixed n, minimize |Rₙ(x)| 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 |Rₙ(x)| to less than 10⁻⁶. Graph the function N(x) on the interval |x|<1.

Accepted Answer

Step 1: Understand the definitions given. The function is defined as the infinite geometric series $$f(x) = \sum_{k=0}^\infty x^k = \frac{1}{1-x}$$ for $$|x| \u003c 1. $$The partial sum up to $$n$$ terms is $$S_n(x) = \sum_{k=0}^{n-1} x^k. $$The remainder after $$n$$ terms is $$R_n(x) = f(x) - S_n(x). $$Step 2: To show that $$R_n(x) = \frac{x^n}{1-x}$$, start from the formula for the partial sum of a geometric series: $$S_n(x) = \frac{1 - x^n}{1 - x}. $$Then express the remainder as $$R_n(x) = f(x) - S_n(x) = \frac{1}{1-x} - \frac{1 - x^n}{1 - x}. $$Simplify this expression to isolate $$R_n(x). $$Step 3: For the graphing part, consider the function $$R_n(x) = \frac{x^n}{1-x} on $$the interval $$|x| \u003c 1$$ for $$n=1, 2, 3. $$Plot these functions to observe how the remainder behaves as $$x$$ varies. Pay attention to the magnitude $$|R_n(x)|$$ and note where it reaches its maximum and minimum values within the interval. Step 4: To minimize $$|R_n(x)|$$ for fixed $$n$$, analyze the function $$|R_n(x)| = \left| \frac{x^n}{1-x} \right|. $$Consider taking the derivative with respect to $$x$$ and setting it to zero to find critical points. Check these points within the domain $$|x| \u003c 1 to $$determine where the minimum occurs. Compare these results with the observations from the graph. Step 5: For the function $$N(x)$$, which is the number of terms needed to make $$|R_n(x)| \u003c 10$$^{-6}$$, set up the inequality $$\left| \frac{x^n}{1-x} \right| \u003c $$10^{-6}. $$Solve for $$n in $$terms of $$x$$, which will give $$N(x). $$Then graph $$N(x)$$ over the interval $$|x| \u003c 1 to $$visualize how many terms are required for different values of $$x.$$

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Key Concepts

Geometric Series and Partial Sums

Remainder (Error) of a Power Series Approximation

Minimization and Graphical Analysis of Functions