<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!

a n = n 2 / 2 n

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

Got questions? Get instant answers now!

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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
what king of growth are you checking .?
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
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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
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
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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?
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?