# 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.

find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
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can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
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Abhi
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Abhi
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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