# Untitled Page 32

- Page ID
- 148374

## 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+1/2^{2}+1/3^{2}+…

(b) 1/lnln 3-1/lnln 6+1/ln ln 9-1/lnln 12+…

(c)

(d)

(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 sin^{2} *x*+cos^{2} *x*=1 holds
for terms up to order *x*^{2}.

**7**.
In classical physics, the kinetic energy *K* of an object of mass *m* moving at velocity *v*
is given by K=^{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

where *c* is the speed of light. The fact that it diverges as *v* → *c* 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 28^{1/3}=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 log_{2} *x* around *x*=1, and use it to evaluate
log_{2} 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}

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*=4.11*m*/*d*^{2}. Cf. problem 20, p. 49.)
Evaluate the constant and linear terms of the Taylor series for the function *E*(*b*).

**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

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*+sinsinsin *x*+… converge?
(solution in the pdf version of the book)

**14**.
Evaluate

(solution in the pdf version of the book)

**15**.
Evaluate

to six decimal places.

**16**.
Euler was the first to prove

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

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

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 *x*=π^{2},
4π^{2}, 9π^{2}, ..., 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

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 *x*^{2} term and prove the result claimed for the sum of the inverse
fourth powers of the integers. (The sums with odd exponents >= 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

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

**18**.
Evaluate

where *n* is an integer.

**19**.
Determine the convergence of the series

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

**20**.
Determine the convergence of the series

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

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)

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.

### Footnotes

*The Physics Teacher*, 43 (2005) 432.