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We illustrate [link] in [link] . In particular, by representing the remainder R N = a N + 1 + a N + 2 + a N + 3 + as the sum of areas of rectangles, we see that the area of those rectangles is bounded above by N f ( x ) d x and bounded below by N + 1 f ( x ) d x . In other words,

R N = a N + 1 + a N + 2 + a N + 3 + > N + 1 f ( x ) d x

and

R N = a N + 1 + a N + 2 + a N + 3 + < N f ( x ) d x .

We conclude that

N + 1 f ( x ) d x < R N < N f ( x ) d x .

Since

n = 1 a n = S N + R N ,

where S N is the N th partial sum, we conclude that

S N + N + 1 f ( x ) d x < n = 1 a n < S N + N f ( x ) d x .
This shows two graphs side by side of the same decreasing concave up function y = f(x) that approaches the x axis in quadrant 1. Rectangles are drawn with a base of 1 over the intervals N through N + 4. The heights of the rectangles in the first graph are determined by the value of the function at the right endpoints of the bases, and those in the second graph are determined by the value at the left endpoints of the bases. The areas of the rectangles are marked: a_(N + 1), a_(N + 2), through a_(N + 4).
Given a continuous, positive, decreasing function f and a sequence of positive terms a n such that a n = f ( n ) for all positive integers n , (a) the areas a N + 1 + a N + 2 + a N + 3 + < N f ( x ) d x , or (b) the areas a N + 1 + a N + 2 + a N + 3 + > N + 1 f ( x ) d x . Therefore, the integral is either an overestimate or an underestimate of the error.

Estimating the value of a series

Consider the series n = 1 1 / n 3 .

  1. Calculate S 10 = n = 1 10 1 / n 3 and estimate the error.
  2. Determine the least value of N necessary such that S N will estimate n = 1 1 / n 3 to within 0.001 .
  1. Using a calculating utility, we have
    S 10 = 1 + 1 2 3 + 1 3 3 + 1 4 3 + + 1 10 3 1.19753 .

    By the remainder estimate, we know
    R N < N 1 x 3 d x .

    We have
    10 1 x 3 d x = lim b 10 b 1 x 3 d x = lim b [ 1 2 x 2 ] N b = lim b [ 1 2 b 2 + 1 2 N 2 ] = 1 2 N 2 .

    Therefore, the error is R 10 < 1 / 2 ( 10 ) 2 = 0.005 .
  2. Find N such that R N < 0.001 . In part a. we showed that R N < 1 / 2 N 2 . Therefore, the remainder R N < 0.001 as long as 1 / 2 N 2 < 0.001 . That is, we need 2 N 2 > 1000 . Solving this inequality for N , we see that we need N > 22.36 . To ensure that the remainder is within the desired amount, we need to round up to the nearest integer. Therefore, the minimum necessary value is N = 23 .
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For n = 1 1 n 4 , calculate S 5 and estimate the error R 5 .

S 5 1.09035 , R 5 < 0.00267

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Key concepts

  • If lim n a n 0 , then the series n = 1 a n diverges.
  • If lim n a n = 0 , the series n = 1 a n may converge or diverge.
  • If n = 1 a n is a series with positive terms a n and f is a continuous, decreasing function such that f ( n ) = a n for all positive integers n , then
    n = 1 a n and 1 f ( x ) d x

    either both converge or both diverge. Furthermore, if n = 1 a n converges, then the N th partial sum approximation S N is accurate up to an error R N where N + 1 f ( x ) d x < R N < N f ( x ) d x .
  • The p -series n = 1 1 / n p converges if p > 1 and diverges if p 1 .

Key equations

  • Divergence test
    If a n 0 as n , n = 1 a n diverges .
  • p -series
    n = 1 1 n p { converges if p > 1 diverges if p 1
  • Remainder estimate from the integral test
    N + 1 f ( x ) d x < R N < N f ( x ) d x

For each of the following sequences, if the divergence test applies, either state that lim n a n does not exist or find lim n a n . If the divergence test does not apply, state why.

a n = n 5 n 2 3

lim n a n = 0 . Divergence test does not apply.

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a n = ( 2 n + 1 ) ( n 1 ) ( n + 1 ) 2

lim n a n = 2 . Series diverges.

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a n = ( 2 n + 1 ) 2 n ( 3 n 2 + 1 ) n

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a n = 2 n 3 n / 2

lim n a n = (does not exist). Series diverges.

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a n = 2 n + 3 n 10 n / 2

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a n = e −2 / n

lim n a n = 1 . Series diverges.

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a n = tan n

lim n a n does not exist. Series diverges.

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a n = 1 cos 2 ( 1 / n ) sin 2 ( 2 / n )

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a n = ( 1 1 n ) 2 n

lim n a n = 1 / e 2 . Series diverges.

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a n = ( ln n ) 2 n

lim n a n = 0 . Divergence test does not apply.

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State whether the given p -series converges.

n = 1 1 n n

Series converges, p > 1 .

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n = 1 1 n 4 3

Series converges, p = 4 / 3 > 1 .

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Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
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absolutely yes
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for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
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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
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in general
s.
Graphene has a hexagonal structure
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Practice Key Terms 4

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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