Notes: `1/4+2/7+3/12+...+n/(n^2+3)+...` Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the...

Saturday, 19 April 2014

$\displaystyle\frac{{1}}{{4}}+\frac{{2}}{{7}}+\frac{{3}}{{12}}+\ldots+\frac{{n}}{{{{n}}^{{2}}+{3}}}+\ldots$ Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the...

$\displaystyle\frac{{1}}{{4}}+\frac{{2}}{{7}}+\frac{{3}}{{12}}+\ldots..+\frac{{n}}{{{{n}}^{{2}}+{3}}}+\ldots\ldots..$

We can write the series as $\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{n}}{{{{n}}^{{2}}+{3}}}$

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

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

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

Refer to the attached graph of the function. From the graph we observe that the function is positive, continuous and decreasing on...

$\displaystyle\frac{{1}}{{4}}+\frac{{2}}{{7}}+\frac{{3}}{{12}}+\ldots..+\frac{{n}}{{{{n}}^{{2}}+{3}}}+\ldots\ldots..$

We can write the series as $\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{n}}{{{{n}}^{{2}}+{3}}}$

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

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

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

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

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

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

Let's first evaluate the indefinite integral $\displaystyle\int\frac{{x}}{{{{x}}^{{2}}+{3}}}{\left.{d}{x}\right.}$

Apply integral substitution: $\displaystyle{u}={{x}}^{{2}}+{3}$

$\displaystyle\Rightarrow{d}{u}={2}{x}{\left.{d}{x}\right.}$

$\displaystyle\int\frac{{x}}{{{{x}}^{{2}}+{3}}}{\left.{d}{x}\right.}=\int\frac{{1}}{{u}}\frac{{{d}{u}}}{{2}}$

Take the constant out and use the common integral: $\displaystyle\int\frac{{1}}{{x}}{\left.{d}{x}\right.}={\ln}{\left|{x}\right|}$

$\displaystyle=\frac{{1}}{{2}}{\ln}{\left|{u}\right|}+{C}$ where C is a constant

Substitute back $\displaystyle{u}={{x}}^{{2}}+{3}$

$\displaystyle=\frac{{1}}{{2}}{\ln}{\left|{{x}}^{{2}}+{3}\right|}+{C}$

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

$\displaystyle=\lim_{{{b}\to\infty}}\frac{{1}}{{2}}{\ln}{\left|{{b}}^{{2}}+{3}\right|}-\frac{{1}}{{2}}{\ln}{\left|{{1}}^{{2}}+{3}\right|}$

$\displaystyle=\infty-\frac{{1}}{{2}}{\ln{{4}}}$

$\displaystyle=\infty$

Since the integral $\displaystyle{\int_{{1}}^{\infty}}\frac{{x}}{{{{x}}^{{2}}+{3}}}{\left.{d}{x}\right.}$ diverges, we can conclude from the integral test that the series diverges.

Notes

Saturday, 19 April 2014

$\displaystyle\frac{{1}}{{4}}+\frac{{2}}{{7}}+\frac{{3}}{{12}}+\ldots+\frac{{n}}{{{{n}}^{{2}}+{3}}}+\ldots$ Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the...

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How are race, gender, and class addressed in Oliver Optic's Rich and Humble?

Saturday, 19 April 2014

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

No comments:

Post a Comment

How are race, gender, and class addressed in Oliver Optic&#39;s Rich and Humble?

$\displaystyle\frac{{1}}{{4}}+\frac{{2}}{{7}}+\frac{{3}}{{12}}+\ldots+\frac{{n}}{{{{n}}^{{2}}+{3}}}+\ldots$ Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the...

How are race, gender, and class addressed in Oliver Optic's Rich and Humble?