1 E: Chapter Exercises
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Exercise 1E.1
True or False? In the following exercises, justify your answer with a proof or a counterexample.
1. If the radius of convergence for a power series ∞∑n=0anxn is 5, then the radius of convergence for the series ∞∑n=1nanxn−1 is also 5.
- Answer
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True
2. Power series can be used to show that the derivative of ex is ex. (Hint: Recall that ex=∞∑n=01n!xn.)
3. For small values of x,sinx≈x.
- Answer
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True
4. The radius of convergence for the Maclaurin series of f(x)=3x is 3
Exercise 1E.2
In the following exercises, find the radius of convergence and the interval of convergence for the given series.
1. ∞∑n=0n2(x−1)n
- Answer
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ROC: 1; IOC: (0,2)
2. ∞∑n=0xnnn
3. ∞∑n=03nxn12n
- Answer
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ROC: 12; IOC: (−16,8)
4. ∞∑n=02nen(x−e)n
Exercise 1E.3
In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.
1. f(x)=x2x+3
- Answer
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∞∑n=0(−1)n3n+1xn; ROC: 3; IOC: (−3,3)
2. f(x)=8x+22x2−3x+1
Exercise 1E.4
In the following exercises, find the power series for the given function using term-by-term differentiation or integration.
1. f(x)=tan−1(2x)
- Answer
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integration: ∞∑n=0(−1)n2n+1(2x)2n+1
2. f(x)=x(2+x2)2
Exercise 1E.5
In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?
1. f(x)=x3−2x2+4,a=−3
- Answer
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p4(x)=(x+3)3−11(x+3)2+39(x+3)−41; exact
2. f(x)=e1/(4x),a=4
Exercise 1E.6
In the following exercises, find the Maclaurin series for the given function.
1. f(x)=cos(3x)
- Answer
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∞∑n=0(−1)n(3x)2n2n!
2. f(x)=ln(x+1)
Exercise 1E.7
In the following exercises, find the Taylor series at the given value.
1. f(x)=sinx,a=π2
- Answer
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∞∑n=0(−1)n(2n)!(x−π2)2n
2. f(x)=3x,a=1
Exercise 1E.8
In the following exercises, find the Maclaurin series for the given function.
1. f(x)=e−x2−1
- Answer
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∞∑n=1(−1)nn!x2n
2. f(x)=cosx−xsinx
Exercise 1E.9
In the following exercises, find the Maclaurin series for F(x)=∫x0f(t)dt by integrating the Maclaurin series of f(x) term by term.
1. f(x)=sinxx
- Answer
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F(x)=∞∑n=0(−1)n(2n+1)(2n+1)!x2n+1
2. f(x)=1−ex
3. Use power series to prove Euler’s formula: eix=cosx+isinx
Exercise 1E.10
The following exercises consider problems of annuity payments.
1. For annuities with a present value of $1 million, calculate the annual payouts given over 25 years assuming interest rates of 1, and 10
2. A lottery winner has an annuity that has a present value of $10 million. What interest rate would they need to live on perpetual annual payments of $250,000?
- Answer
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2.5
3. Calculate the necessary present value of an annuity in order to support annual payouts of $15,000 given over 25 years assuming interest rates of 1,and 10