# 5.6 Ratio and root tests  (Page 4/8)

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## Series converging to $\pi$ And $1\text{/}\pi$

Dozens of series exist that converge to $\pi$ or an algebraic expression containing $\pi .$ 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 $\pi$ 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
$\pi =4\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{2n-1}=4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\text{⋯}$

was discovered by Gregory and Leibniz in the late $1600\text{s}\text{.}$ This result follows from the Maclaurin series for $f\left(x\right)={\text{tan}}^{-1}x.$ 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 $|{R}_{N}|<0.01\text{?}$ Evaluate ${S}_{N}.$
2. The series
$\begin{array}{cc}\hfill \pi & =6\sum _{n=0}^{\infty }\frac{\left(2n\right)\text{!}}{{2}^{4n+1}{\left(n\text{!}\right)}^{2}\left(2n+1\right)}\hfill \\ & =6\left(\frac{1}{2}+\frac{1}{2·3}{\left(\frac{1}{2}\right)}^{3}+\frac{1·3}{2·4·5}·{\left(\frac{1}{2}\right)}^{5}+\frac{1·3·5}{2·4·6·7}{\left(\frac{1}{2}\right)}^{7}+\text{⋯}\right)\hfill \end{array}$

has been attributed to Newton in the late $1600\text{s}\text{.}$ The proof of this result uses the Maclaurin series for $f\left(x\right)={\text{sin}}^{-1}x.$
1. Prove that the series converges.
2. Evaluate the partial sums ${S}_{n}$ for $n=5,10,20.$
3. Compare ${S}_{n}$ to $\pi$ for $n=5,10,20$ and discuss the number of correct decimal places.
3. The series
$\frac{1}{\pi }=\frac{\sqrt{8}}{9801}\sum _{n=0}^{\infty }\frac{\left(4n\right)\text{!}\left(1103+26390n\right)}{{\left(n\text{!}\right)}^{4}{396}^{4n}}$

was discovered by Ramanujan in the early $1900\text{s}\text{.}$ William Gosper, Jr., used this series to calculate $\pi$ to an accuracy of more than $17$ million digits in the $\text{mid-}1980\text{s}\text{.}$ 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 $\pi$ and $1\text{/}\pi .$
1. Prove that this series converges.
2. Evaluate the first term in this series. Compare this number with the value of $\pi$ 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 $\left(1887\text{–}1920\right)$ 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
$\rho =\underset{n\to \infty }{\text{lim}}|\frac{{a}_{n+1}}{{a}_{n}}|.$

If $\rho <1,$ the series $\sum _{n=1}^{\infty }{a}_{n}$ converges absolutely. If $\rho >1,$ the series diverges. If $\rho =1,$ the test does not provide any information. This test is useful for series whose terms involve factorials.
• For the root test, we consider
$\rho =\underset{n\to \infty }{\text{lim}}\sqrt[n]{|{a}_{n}|}.$

If $\rho <1,$ the series $\sum _{n=1}^{\infty }{a}_{n}$ converges absolutely. If $\rho >1,$ the series diverges. If $\rho =1,$ 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 $p-\text{series,}$ consider one of the comparison tests.

Use the ratio test to determine whether $\sum _{n=1}^{\infty }{a}_{n}$ converges, where ${a}_{n}$ is given in the following problems. State if the ratio test is inconclusive.

${a}_{n}=1\text{/}n\text{!}$

${a}_{n+1}\text{/}{a}_{n}\to 0.$ Converges.

${a}_{n}={10}^{n}\text{/}n\text{!}$

${a}_{n}={n}^{2}\text{/}{2}^{n}$

$\frac{{a}_{n+1}}{{a}_{n}}=\frac{1}{2}{\left(\frac{n+1}{n}\right)}^{2}\to 1\text{/}2<1.$ Converges.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
learn
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
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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
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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?
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?
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