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7.5. Homework Problems

1. Modify the Weierstrass definition of the limit to apply to infinite sequences. (solution in the pdf version of the book)

2. (a) Prove that the infinite series 1-1+1-1+1-1+… does not converge to any limit, using the generalization of the Weierstrass limit found in problem 1.
(b) Criticize the following argument. The series given in part (a) equals zero, because addition is associative, so we can rewrite it as (1-1) + (1-1) + (1-1) +… (solution in the pdf version of the book)

3. Use the integral test to prove the convergence of the geometric series for 0<x<1. (solution in the pdf version of the book)

4. Determine the convergence or divergence of the following series.
(a) 1 + 1/22 + 1/32 + …
(b) 1/ln ln 3 - 1/ln ln 6 + 1/ln ln 9 - 1/ln ln 12 +…
(c)

\begin{align} \frac{1}{\ln 2} + \frac{1}{(\ln 2)(\ln 3)} + \frac{1}{(\ln 2)(\ln 3)(\ln 4)} + ... \end{align}

(d)

\[ \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}} \]

(solution in the pdf version of the book)

5. Give an example of a series for which the ratio test is inconclusive. (solution in the pdf version of the book)

6. Find the Taylor series expansion of cos x around x=0. Check your work by combining the first two terms of this series with the first term of the sine function from Example 85 on page 112 to verify that the trig identity sin2 x + cos2 x = 1 holds for terms up to order x2.

7. In classical physics, the kinetic energy K of an object of mass m moving at velocity v is given by \(K=\frac{1}{2}mv^2\). For example, if a car is to start from a stoplight and then accelerate up to v, this is the theoretical minimum amount of energy that would have to be used up by burning gasoline. (In reality, a car's engine is not 100% efficient, so the amount of gas burned is greater.)

Einstein's theory of relativity states that the correct equation is actually

\[ K = \left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right)mc^2 , \]

where c is the speed of light. The fact that it diverges as vc is interpreted to mean that no object can be accelerated to the speed of light.

Expand K in a Taylor series, and show that the first nonvanishing term is equal to the classical expression. This means that for velocities that are small compared to the speed of light, the classical expression is a good approximation, and Einstein's theory does not contradict any of the prior empirical evidence from which the classical expression was inferred.

8. Expand (1+x)1/3 in a Taylor series around x=0. The value x=28 lies outside this series' radius of convergence, but we can nevertheless use it to extract the cube root of 28 by recognizing that 281/3 = 3(28/27)1/3. Calculate the root to four significant figures of precision, and check it in the obvious way.

9. Find the Taylor series expansion of log2 x around x=1, and use it to evaluate log2 1.0595 to four significant figures of precision. Check your result by using the fact that 1.0595 is approximately the twelfth root of 2. This number is the ratio of the frequencies of two successive notes of the chromatic scale in music, e.g., C and D-flat.

10. In free fall, the acceleration will not be exactly constant, due to air resistance. For example, a skydiver does not speed up indefinitely until opening her chute, but rather approaches a certain maximum velocity at which the upward force of air resistance cancels out the force of gravity. If an object is dropped from a height h, and the time it takes to reach the ground is used to measure the acceleration of gravity, g, then the relative error in the result due to air resistance is [3]

\begin{align} E = \frac{g-g_{vacuum}}{g} & = 1-\frac{2b}{\ln^2\left(e^b+\sqrt{e^{2b}-1}\right)} , \end{align}

where b=h/A, and A is a constant that depends on the size, shape, and mass of the object, and the density of the air. (For a sphere of mass m and diameter d dropping in air, A = 4.11m/d2. Cf. problem 20, p. 49.) Evaluate the constant and linear terms of the Taylor series for the function E(b).

[3] Jan Benacka and Igor Stubna, The Physics Teacher, 43 (2005) 432.

11. (a) Prove that the convergence of an infinite series is unaffected by omitting some initial terms. (b) Similarly, prove that convergence is unaffected by multiplying all the terms by some constant factor.

12. The identity

\[ \int_0^1 x^{-x} dx = \sum_{n=1}^\infty n^{-n} . \]

is known as the “Sophomore's dream,” because at first glance it looks like the kind of plausible but false statement that someone would naively dream up. Verify it numerically by machine computation.

13. Does sin x + sin sin x + sin sin sin x +… converge? (solution in the pdf version of the book)

14. Evaluate

\[ 1+\frac{1}{1+2}+\frac{1}{1+2+3}+... \]

(solution in the pdf version of the book)

15. Evaluate

\[ \sum_{n=0}^\infty \frac{(-1)^n}{n+1+1/n!} \]

to six decimal places.

16. Euler was the first to prove

\[ \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...=\frac{\pi^2}{6} . \]

This problem had defeated other great mathematicians of his time, and was famous enough to be given a special name, the Basel problem. Here we present an argument based closely on Euler's and pose the problem of how to exploit Euler's technique further in order to prove

\[ \frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+... = \frac{\pi^4}{90} . \]

From the Taylor series for the sine function, we find the related series

\[ f(x) = \frac{\sin\sqrt{x}}{\sqrt{x}} = 1 - \frac{x}{3!} + \frac{x^2}{5!} . \]

The partial sums of this series are polynomials that approximate f for small values of x. If such a polynomial were exact rather than approximate, then it would have zeroes at x2, 4π2, 9π2, ..., and we could write it as the product of its linear factors. Euler assumed, without any more rigorous proof, that this factorization procedure could be extended to the infinite series, so that f could be represented as the infinite product

\[ f(x) = \left(1-\frac{x}{\pi^2}\right)\left(1-\frac{x}{4\pi^2}\right)... \]

By multiplying this out and equating its linear term to that of the Taylor series, we find the claimed result.

Extend this procedure to the x2 term and prove the result claimed for the sum of the inverse fourth powers of the integers. (The sums with odd exponents \(\ge 3\) are much harder, and relatively little is known about them. The sum of the inverse cubes is known as Apèry's constant.)

17. Does

\[ \int_0^\infty \sin (x^2)dx \]

converge, or not? (solution in the pdf version of the book)

18. Evaluate

\[ \lim_{n\rightarrow\infty} \cos(\pi\sqrt{n^2-n}) , \]

where n is an integer.

19. Determine the convergence of the series

\[ \sum_{n=0}^{\infty} n^2 2^{-n} , \]

and if it converges, evaluate it. (solution in the pdf version of the book)

20. Determine the convergence of the series

\[ \sum_{n=0}^{\infty} n^2 2^{-n} , \]

and if it converges, evaluate it. (solution in the pdf version of the book)

21. For what integer values of p should we expect the series

\[ \sum_{n=1}^{\infty} \frac{|\cos n|^n}{n^p} \]

to converge? A rigorous proof is very difficult and may even be an open problem, but it is relatively straightforward to give a convincing argument. (solution in the pdf version of the book)