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Series converging to π And 1 / π

Dozens of series exist that converge to π or an algebraic expression containing π . Here we look at several examples and compare their rates of convergence. By rate of convergence, we mean the number of terms necessary for a partial sum to be within a certain amount of the actual value. The series representations of π in the first two examples can be explained using Maclaurin series, which are discussed in the next chapter. The third example relies on material beyond the scope of this text.

  1. The series
    π = 4 n = 1 ( −1 ) n + 1 2 n 1 = 4 4 3 + 4 5 4 7 + 4 9

    was discovered by Gregory and Leibniz in the late 1600 s . This result follows from the Maclaurin series for f ( x ) = tan −1 x . We will discuss this series in the next chapter.
    1. Prove that this series converges.
    2. Evaluate the partial sums S n for n = 10 , 20 , 50 , 100 .
    3. Use the remainder estimate for alternating series to get a bound on the error R n .
    4. What is the smallest value of N that guarantees | R N | < 0.01 ? Evaluate S N .
  2. The series
    π = 6 n = 0 ( 2 n ) ! 2 4 n + 1 ( n ! ) 2 ( 2 n + 1 ) = 6 ( 1 2 + 1 2 · 3 ( 1 2 ) 3 + 1 · 3 2 · 4 · 5 · ( 1 2 ) 5 + 1 · 3 · 5 2 · 4 · 6 · 7 ( 1 2 ) 7 + )

    has been attributed to Newton in the late 1600 s . The proof of this result uses the Maclaurin series for f ( x ) = sin −1 x .
    1. Prove that the series converges.
    2. Evaluate the partial sums S n for n = 5 , 10 , 20 .
    3. Compare S n to π for n = 5 , 10 , 20 and discuss the number of correct decimal places.
  3. The series
    1 π = 8 9801 n = 0 ( 4 n ) ! ( 1103 + 26390 n ) ( n ! ) 4 396 4 n

    was discovered by Ramanujan in the early 1900 s . William Gosper, Jr., used this series to calculate π to an accuracy of more than 17 million digits in the mid- 1980 s . At the time, that was a world record. Since that time, this series and others by Ramanujan have led mathematicians to find many other series representations for π and 1 / π .
    1. Prove that this series converges.
    2. Evaluate the first term in this series. Compare this number with the value of π from a calculating utility. To how many decimal places do these two numbers agree? What if we add the first two terms in the series?
    3. Investigate the life of Srinivasa Ramanujan ( 1887 1920 ) and write a brief summary. Ramanujan is one of the most fascinating stories in the history of mathematics. He was basically self-taught, with no formal training in mathematics, yet he contributed in highly original ways to many advanced areas of mathematics.

Key concepts

  • For the ratio test, we consider
    ρ = lim n | a n + 1 a n | .

    If ρ < 1 , the series n = 1 a n converges absolutely. If ρ > 1 , the series diverges. If ρ = 1 , the test does not provide any information. This test is useful for series whose terms involve factorials.
  • For the root test, we consider
    ρ = lim n | a n | n .

    If ρ < 1 , the series n = 1 a n converges absolutely. If ρ > 1 , the series diverges. If ρ = 1 , the test does not provide any information. The root test is useful for series whose terms involve powers.
  • For a series that is similar to a geometric series or p series, consider one of the comparison tests.

Use the ratio test to determine whether n = 1 a n converges, where a n is given in the following problems. State if the ratio test is inconclusive.

a n = 1 / n !

a n + 1 / a n 0 . Converges.

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a n = n 2 / 2 n

a n + 1 a n = 1 2 ( n + 1 n ) 2 1 / 2 < 1 . Converges.

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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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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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.
yes that's correct
I think
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is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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scanning tunneling microscope
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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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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