New Notes Blog: `sum_(n=1)^oo 2/(3n+5)` Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence...

Friday, 11 September 2015

$\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{2}}{{{3}{n}+{5}}}$ Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence...

$\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{2}}{{{3}{n}+{5}}}$

The Integral test is applicable if f is positive, continuous and decreasing function on the infinite interval $\displaystyle{\left[{k},\infty\right)}$ where $\displaystyle{k}\ge{1}$ and $\displaystyle{a}_{{n}}={f{{\left({x}\right)}}}$ . Then the series converges or diverges if and only if the improper integral $\displaystyle{\int_{{k}}^{\infty}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ converges or diverges.

For the given series $\displaystyle{a}_{{n}}=\frac{{2}}{{{3}{n}+{5}}}$

Consider $\displaystyle{f{{\left({x}\right)}}}=\frac{{2}}{{{3}{x}+{5}}}$

Refer to the attached graph of the function. It is positive and continuous on the interval $\displaystyle{\left[{1},\infty\right)}$

We can determine whether $\displaystyle{f{{\left({x}\right)}}}$ is decreasing by finding the derivative $\displaystyle{f{'}}{\left({x}\right)}$ , $\displaystyle{f{'}}{\left({x}\right)}<{0}$ for $\displaystyle{x}\ge{1}$

...

$\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{2}}{{{3}{n}+{5}}}$

For the given series $\displaystyle{a}_{{n}}=\frac{{2}}{{{3}{n}+{5}}}$

Consider $\displaystyle{f{{\left({x}\right)}}}=\frac{{2}}{{{3}{x}+{5}}}$

Refer to the attached graph of the function. It is positive and continuous on the interval $\displaystyle{\left[{1},\infty\right)}$

$\displaystyle{f{'}}{\left({x}\right)}={2}{\left(-{1}\right)}{{\left({3}{x}+{5}\right)}}^{{-{2}}}{\left({3}\right)}$

$\displaystyle{f{'}}{\left({x}\right)}=-\frac{{6}}{{{\left({3}{x}+{5}\right)}}^{{2}}}$

So, $\displaystyle{f{'}}{\left({x}\right)}<{0}$

We can apply integral test as the function satisfies the conditions for the integral test.

Now let's determine whether the corresponding improper integral $\displaystyle{\int_{{1}}^{\infty}}\frac{{2}}{{{3}{x}+{5}}}{\left.{d}{x}\right.}$ converges or diverges as:

$\displaystyle{\int_{{1}}^{\infty}}\frac{{2}}{{{3}{x}+{5}}}{\left.{d}{x}\right.}=\lim_{{{b}\to\infty}}{\int_{{1}}^{{b}}}\frac{{2}}{{{3}{x}+{5}}}{\left.{d}{x}\right.}$

$\displaystyle=\lim_{{{b}\to\infty}}{{\left[\frac{{2}}{{3}}{\ln}{\left|{3}{x}+{5}\right|}\right]}_{{1}}^{{b}}}$

$\displaystyle=\lim_{{{b}\to\infty}}\frac{{2}}{{3}}{\left[{\ln}{\left|{3}{b}+{5}\right|}-{\ln}{\left|{3}+{5}\right|}\right]}$

$\displaystyle=\frac{{2}}{{3}}{\left[\infty-{\ln{{8}}}\right]}$

$\displaystyle=\infty$ which implies that the integral diverges.

Therefore the series $\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{2}}{{{3}{n}+{5}}}$ must also diverge.

New Notes Blog

Friday, 11 September 2015

$\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{2}}{{{3}{n}+{5}}}$ Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence...

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Friday, 11 September 2015

Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence...

No comments:

Post a Comment

In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{2}}{{{3}{n}+{5}}}$ Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence...

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?