The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications

ddaeeric 2021-09-30 Answered

The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if $ \(\lim_{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}<1 / 2\) then \(\displaystyle\sum{a}_{{{n}}}\) converges, while if \(\lim_{n \rightarrow \infty} \frac{a_{2 n+1}}{a_{n}}>1 / 2\) then \(\displaystyle\sum{a}_{{{n}}}\)  diverges. Let \(\displaystyle{a}_{{{n}}}={\frac{{{n}^{{{\ln{{n}}}}}}}{{{\left({\ln{{n}}}\right)}^{{{n}}}}}}\). Show that \(\frac{a_{2n}}{a_{n}} \rightarrow 0\) as \(\displaystyle{n}\rightarrow\infty\)

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Expert Answer

Jayden-James Duffy
Answered 2021-10-01 Author has 17372 answers
Proved, \(\displaystyle\lim_{{{n}\rightarrow\infty}}{\frac{{{a}_{{{2}{n}}}}}{{{a}_{{{n}}}}}}={0}\)
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Relevant Questions

asked 2021-06-16

The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if $ \(\displaystyle\lim{\left\lbrace{n}\rightarrow\infty\right\rbrace}{\frac{{{a}{\left\lbrace{2}{n}\right\rbrace}}}{{{a}_{{{n}}}}}}{<}\frac{{1}}{{2}}\) then \(\displaystyle\sum{a}_{{{n}}}\) converges, while if \(\displaystyle\lim{\left\lbrace{n}\rightarrow\infty\right\rbrace}{\frac{{{a}{\left\lbrace{2}{n}+{1}\right\rbrace}}}{{{a}_{{{n}}}}}}{>}\frac{{1}}{{2}}\) then \(\displaystyle\sum{a}_{{{n}}}\) $ diverges. Let \(\displaystyle{a}_{{{n}}}={\frac{{{n}^{{{\ln{{n}}}}}}}{{{\left({\ln{{n}}}\right)}^{{{n}}}}}}\). Show that \(\frac{a_{2n}}{a_{n}} \rightarrow 0\) as \(\displaystyle{n}\rightarrow\infty\)

asked 2021-05-05

The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if $$ \(\displaystyle\lim{\left\lbrace{n}\rightarrow\infty\right\rbrace}{\frac{{{a}{\left\lbrace{2}{n}\right\rbrace}}}{{{a}_{{{n}}}}}}{<}\frac{{1}}{{2}}\) then \(\sum a_{n} \)converges,while if \(\displaystyle\lim{\left\lbrace{n}\rightarrow\infty\right\rbrace}{\frac{{{a}{\left\lbrace{2}{n}+{1}\right\rbrace}}}{{{a}_{{{n}}}}}}{>}\frac{{1}}{{2}}\), then \(\sum a_{n}\) diverges.

Let \(\displaystyle{a}_{{{n}}}={\frac{{{1}}}{{{1}+{x}}}}{\frac{{{2}}}{{{2}+{x}}}}\ldots{\frac{{{n}}}{{{n}+{x}}}}{\frac{{{1}}}{{{n}}}}={\frac{{{\left({n}-{1}\right)}!}}{{{\left({1}+{x}\right)}{\left({2}+{x}\right)}\ldots{\left({n}+{x}\right)}}}}\).

Show that \(\frac{a_{2 n}}{a_{n}} \leq \frac{e^{-x / 2}}{2}\) . For which x > 0 does the generalized ratio test imply convergence of \(\sum_{n=1}^\infty a_{n}\)?

asked 2021-08-14

The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence.

The test states that if

 \(\lim_{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}<1 / 2\)

then \(\sum a_{n}\) converges,while if

\(\lim_{n \rightarrow \infty} \frac{a_{2 n+1}}{a_{n}}>1 / 2\),

then \(\sum a_{n}\)  diverges. Let \(\displaystyle{a}_{{{n}}}={\frac{{{1}}}{{{1}+{x}}}}{\frac{{{2}}}{{{2}+{x}}}}\ldots{\frac{{{n}}}{{{n}+{x}}}}{\frac{{{1}}}{{{n}}}}={\frac{{{\left({n}-{1}\right)}!}}{{{\left({1}+{x}\right)}{\left({2}+{x}\right)}\ldots{\left({n}+{x}\right)}}}}\).

Show that \(a_{2 n}/a_{n} \leq e^{-x/2}/ 2\)  .

For which x > 0 does the generalized ratio test imply convergence of \(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{a}_{{{n}}}\)?

asked 2021-10-07

Bessel functions often arise in advanced engineering analyses such as the study of electric fields. Here are some selected values for the zero-order Bessel function of the first kind.
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Estimate \(J_1(2.1)\) using third- and fourth-order interpolating polynomials. Determine the percent relative error for each case based on the true value, which can be determined with MATLAB’s built-in function besselj.

asked 2021-10-03

Bessel functions often arise in advanced engineering and scientific analyses such as the study of electric fields. These functions are usually not amenable to straightforward evaluation and, therefore, are often compiled in standard mathematical tables.
\(\begin{array}\\ x&1.8&2.0&2.2&2.4&2.6\\ J_1(x)&0.5815&0.5767&0.5560&0.5202&0.4708 \end{array}\)
Estimate \(J_1(2.1)\), (a) using an interpolating polynomial and (b) using cubic splines. Note that the true value is 0.5683

asked 2021-10-24
Test the series for convergence or divergence.
\(\displaystyle{\sum_{{{n}={0}}}^{\infty}}{\frac{{{\left(-{1}\right)}^{{{n}+{1}}}}}{{\sqrt{{{n}+{4}}}}}}\)
asked 2021-11-12
Use the Integral Test to determine whether the series is convergent or divergent.
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\frac{{{n}}}{{{n}^{{2}}+{1}}}}\)

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