Jayden-James Duffy

Answered 2021-10-01
Author has **17372** answers

asked 2021-06-16

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

\(\begin{array}\\ x&1.8&2.0&2.2&2.4&2.6\\ J_1(x)&0.5815&0.5767&0.5560&0.5202&0.4708 \end{array}\)

Estimate \(J_1(2.1)\) 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

\(\begin{array}\\ x&1.8&2.0&2.2&2.4&2.6\\ J_1(x)&0.5815&0.5767&0.5560&0.5202&0.4708 \end{array}\)

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.

\(\displaystyle{\sum_{{{n}={0}}}^{\infty}}{\frac{{{\left(-{1}\right)}^{{{n}+{1}}}}}{{\sqrt{{{n}+{4}}}}}}\)

\(\displaystyle{\sum_{{{n}={0}}}^{\infty}}{\frac{{{\left(-{1}\right)}^{{{n}+{1}}}}}{{\sqrt{{{n}+{4}}}}}}\)

asked 2021-11-12

Use the Integral Test to determine whether the series is convergent or divergent.

\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\frac{{{n}}}{{{n}^{{2}}+{1}}}}\)

\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\frac{{{n}}}{{{n}^{{2}}+{1}}}}\)