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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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?
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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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