<< Chapter < Page Chapter >> Page >

Suppose that a n > 0 for all n and that n = 1 a n converges. Suppose that b n is an arbitrary sequence of zeros and ones. Does n = 1 a n b n necessarily converge?

Got questions? Get instant answers now!

Suppose that a n > 0 for all n and that n = 1 a n diverges. Suppose that b n is an arbitrary sequence of zeros and ones with infinitely many terms equal to one. Does n = 1 a n b n necessarily diverge?

No. n = 1 1 / n diverges. Let b k = 0 unless k = n 2 for some n . Then k b k / k = 1 / k 2 converges.

Got questions? Get instant answers now!

Complete the details of the following argument: If n = 1 1 n converges to a finite sum s , then 1 2 s = 1 2 + 1 4 + 1 6 + and s 1 2 s = 1 + 1 3 + 1 5 + . Why does this lead to a contradiction?

Got questions? Get instant answers now!

Show that if a n 0 and n = 1 a 2 n converges, then n = 1 sin 2 ( a n ) converges.

| sin t | | t | , so the result follows from the comparison test.

Got questions? Get instant answers now!

Suppose that a n / b n 0 in the comparison test, where a n 0 and b n 0. Prove that if b n converges, then a n converges.

Got questions? Get instant answers now!

Let b n be an infinite sequence of zeros and ones. What is the largest possible value of x = n = 1 b n / 2 n ?

By the comparison test, x = n = 1 b n / 2 n n = 1 1 / 2 n = 1.

Got questions? Get instant answers now!

Let d n be an infinite sequence of digits, meaning d n takes values in { 0 , 1 ,… , 9 } . What is the largest possible value of x = n = 1 d n / 10 n that converges?

Got questions? Get instant answers now!

Explain why, if x > 1 / 2 , then x cannot be written x = n = 2 b n 2 n ( b n = 0 or 1 , b 1 = 0 ) .

If b 1 = 0 , then, by comparison, x n = 2 1 / 2 n = 1 / 2 .

Got questions? Get instant answers now!

[T] Evelyn has a perfect balancing scale, an unlimited number of 1 -kg weights, and one each of 1 / 2 -kg , 1 / 4 -kg , 1 / 8 -kg , and so on weights. She wishes to weigh a meteorite of unspecified origin to arbitrary precision. Assuming the scale is big enough, can she do it? What does this have to do with infinite series?

Got questions? Get instant answers now!

[T] Robert wants to know his body mass to arbitrary precision. He has a big balancing scale that works perfectly, an unlimited collection of 1 -kg weights, and nine each of 0.1 -kg, 0.01 -kg , 0.001 -kg, and so on weights. Assuming the scale is big enough, can he do this? What does this have to do with infinite series?

Yes. Keep adding 1 -kg weights until the balance tips to the side with the weights. If it balances perfectly, with Robert standing on the other side, stop. Otherwise, remove one of the 1 -kg weights, and add 0.1 -kg weights one at a time. If it balances after adding some of these, stop. Otherwise if it tips to the weights, remove the last 0.1 -kg weight. Start adding 0.01 -kg weights. If it balances, stop. If it tips to the side with the weights, remove the last 0.01 -kg weight that was added. Continue in this way for the 0.001 -kg weights, and so on. After a finite number of steps, one has a finite series of the form A + n = 1 N s n / 10 n where A is the number of full kg weights and d n is the number of 1 / 10 n -kg weights that were added. If at some state this series is Robert’s exact weight, the process will stop. Otherwise it represents the N th partial sum of an infinite series that gives Robert’s exact weight, and the error of this sum is at most 1 / 10 N .

Got questions? Get instant answers now!

The series n = 1 1 2 n is half the harmonic series and hence diverges. It is obtained from the harmonic series by deleting all terms in which n is odd. Let m > 1 be fixed. Show, more generally, that deleting all terms 1 / n where n = m k for some integer k also results in a divergent series.

Got questions? Get instant answers now!

In view of the previous exercise, it may be surprising that a subseries of the harmonic series in which about one in every five terms is deleted might converge. A depleted harmonic series is a series obtained from n = 1 1 n by removing any term 1 / n if a given digit, say 9 , appears in the decimal expansion of n . Argue that this depleted harmonic series converges by answering the following questions.

  1. How many whole numbers n have d digits?
  2. How many d -digit whole numbers h ( d ) . do not contain 9 as one or more of their digits?
  3. What is the smallest d -digit number m ( d ) ?
  4. Explain why the deleted harmonic series is bounded by d = 1 h ( d ) m ( d ) .
  5. Show that d = 1 h ( d ) m ( d ) converges.

a. 10 d 10 d 1 < 10 d b. h ( d ) < 9 d c. m ( d ) = 10 d 1 + 1 d. Group the terms in the deleted harmonic series together by number of digits. h ( d ) bounds the number of terms, and each term is at most 1 / m ( d ) . d = 1 h ( d ) / m ( d ) d = 1 9 d / ( 10 ) d 1 90. One can actually use comparison to estimate the value to smaller than 80 . The actual value is smaller than 23 .

Got questions? Get instant answers now!

Suppose that a sequence of numbers a n > 0 has the property that a 1 = 1 and a n + 1 = 1 n + 1 S n , where S n = a 1 + + a n . Can you determine whether n = 1 a n converges? ( Hint: S n is monotone.)

Got questions? Get instant answers now!

Suppose that a sequence of numbers a n > 0 has the property that a 1 = 1 and a n + 1 = 1 ( n + 1 ) 2 S n , where S n = a 1 + + a n . Can you determine whether n = 1 a n converges? ( Hint: S 2 = a 2 + a 1 = a 2 + S 1 = a 2 + 1 = 1 + 1 / 4 = ( 1 + 1 / 4 ) S 1 , S 3 = 1 3 2 S 2 + S 2 = ( 1 + 1 / 9 ) S 2 = ( 1 + 1 / 9 ) ( 1 + 1 / 4 ) S 1 , etc. Look at ln ( S n ) , and use ln ( 1 + t ) t , t > 0 . )

Continuing the hint gives S N = ( 1 + 1 / N 2 ) ( 1 + 1 / ( N 1 ) 2 ( 1 + 1 / 4 ) ) . Then ln ( S N ) = ln ( 1 + 1 / N 2 ) + ln ( 1 + 1 / ( N 1 ) 2 ) + + ln ( 1 + 1 / 4 ) . Since ln ( 1 + t ) is bounded by a constant times t , when 0 < t < 1 one has ln ( S N ) C n = 1 N 1 n 2 , which converges by comparison to the p -series for p = 2 .

Got questions? Get instant answers now!

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Abdul Reply
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 2' conversation and receive update notifications?

Ask