No, he ratio test does NOT imply the convergence for \(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{a}_{{{n}}}\).

asked 2021-09-30

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 2021-06-16

asked 2021-06-11

i.If \(a_{n}\ \text{and}\ f(n)\) satisfy the requirements of the integral test, then \(\sum_{n=1}^{\infty} a_n = \int_1^{\infty} f(x)dx\)

ii. The series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\ \text{converges if}\ p > 1\ \text{and diverges if}\ p \leq 1\).

iii. The integral test does not apply to divergent sequences.

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 2022-01-18

Test the condition for convergence of

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

and find the sum if it exists.

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

and find the sum if it exists.

asked 2021-05-27

Evaluate the indefinite integral as a power series.

\(\int \frac{\tan^{-1}x}{x}dx\)

\(f(x)=C+\sum_{n=0}^\infty\left( \dots \right)\)

What is the radius of convergence R?

\(\int \frac{\tan^{-1}x}{x}dx\)

\(f(x)=C+\sum_{n=0}^\infty\left( \dots \right)\)

What is the radius of convergence R?