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

Friday, 11 October 2013

$\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{{4}{n}}}{{{2}{n}+{1}}}$ Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the...

$\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{{4}{n}}}{{{2}{n}+{1}}}$

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 $\displaystyle{a}_{{n}}=\frac{{{4}{n}}}{{{2}{n}+{1}}}$

Consider $\displaystyle{f{{\left({x}\right)}}}=\frac{{{4}{x}}}{{{2}{x}+{1}}}$

Refer to the attached graph of the function. From the graph we observe that the function is positive and continuous. However it is not decreasing on the interval $\displaystyle{\left[{1},\infty\right)}$

We...

$\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{{4}{n}}}{{{2}{n}+{1}}}$

For the given series $\displaystyle{a}_{{n}}=\frac{{{4}{n}}}{{{2}{n}+{1}}}$

Consider $\displaystyle{f{{\left({x}\right)}}}=\frac{{{4}{x}}}{{{2}{x}+{1}}}$

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

Let's find the derivative by the quotient rule:

$\displaystyle{f{{\left({x}\right)}}}=\frac{{{4}{x}}}{{{2}{x}+{1}}}$

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{{\left({2}{x}+{1}\right)}\frac{{d}}{{\left.{d}{x}}}{\left({4}{x}\right)}-{\left({4}{x}\right)}\frac{{d}}{{\left.{d}{x}}}{\left({2}{x}+{1}\right)}}}{{{\left({2}{x}+{1}\right)}}^{{2}}}$

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{{\left({2}{x}+{1}\right)}{\left({4}\right)}-{\left({4}{x}\right)}{\left({2}\right)}}}{{{\left({2}{x}+{1}\right)}}^{{2}}}$

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{{8}{x}+{4}-{8}{x}}}{{{\left({2}{x}+{1}\right)}}^{{2}}}$

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{4}}{{{\left({2}{x}+{1}\right)}}^{{2}}}$

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

which implies that the function is not decreasing.

Since the function does not satisfies the conditions for the integral test, we can not apply integral test.

Notes

Friday, 11 October 2013

$\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{{4}{n}}}{{{2}{n}+{1}}}$ 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?

Friday, 11 October 2013

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?

$\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{{4}{n}}}{{{2}{n}+{1}}}$ 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?