Notes: `1/2 + 1/5 + 1/10+1/17+1/26+...` Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the...

Wednesday, 4 February 2015

$\displaystyle\frac{{1}}{{2}}+\frac{{1}}{{5}}+\frac{{1}}{{10}}+\frac{{1}}{{17}}+\frac{{1}}{{26}}+\ldots$ Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the...

$\displaystyle\frac{{1}}{{2}}+\frac{{1}}{{5}}+\frac{{1}}{{10}}+\frac{{1}}{{17}}+\frac{{1}}{{26}}+\ldots\ldots\ldots\ldots$

The series can be written as,

$\displaystyle\frac{{1}}{{{{1}}^{{2}}+{1}}}+\frac{{1}}{{{{2}}^{{2}}+{1}}}+\frac{{1}}{{{{3}}^{{2}}+{1}}}+\frac{{1}}{{{{4}}^{{2}}+{1}}}+\frac{{1}}{{{{5}}^{{2}}+{1}}}+\ldots\ldots\ldots.$

So based on the above pattern we can write the series as,

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

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

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

Graph of the function is attached. From the...

$\displaystyle\frac{{1}}{{2}}+\frac{{1}}{{5}}+\frac{{1}}{{10}}+\frac{{1}}{{17}}+\frac{{1}}{{26}}+\ldots\ldots\ldots\ldots$

The series can be written as,

$\displaystyle\frac{{1}}{{{{1}}^{{2}}+{1}}}+\frac{{1}}{{{{2}}^{{2}}+{1}}}+\frac{{1}}{{{{3}}^{{2}}+{1}}}+\frac{{1}}{{{{4}}^{{2}}+{1}}}+\frac{{1}}{{{{5}}^{{2}}+{1}}}+\ldots\ldots\ldots.$

So based on the above pattern we can write the series as,

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

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

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

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

Since the function satisfies the conditions for the integral test, we can apply integral test.

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

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

Use the common integral: $\displaystyle\int\frac{{1}}{{{{x}}^{{2}}+{1}}}{\left.{d}{x}\right.}={\arctan{{\left({x}\right)}}}$

$\displaystyle=\lim_{{{b}\to\infty}}{{\left[{\arctan{{\left({x}\right)}}}\right]}_{{1}}^{{b}}}$

$\displaystyle=\lim_{{{b}\to\infty}}{\left[{\arctan{{\left({b}\right)}}}-{\arctan{{\left({1}\right)}}}\right]}$

$\displaystyle=\lim_{{{b}\to\infty}}{\arctan{{\left({b}\right)}}}-{\arctan{{\left({1}\right)}}}$

$\displaystyle=\frac{\pi}{{2}}-\frac{\pi}{{4}}$

$\displaystyle=\frac{\pi}{{4}}$

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

Notes

Wednesday, 4 February 2015

$\displaystyle\frac{{1}}{{2}}+\frac{{1}}{{5}}+\frac{{1}}{{10}}+\frac{{1}}{{17}}+\frac{{1}}{{26}}+\ldots$ 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?

Wednesday, 4 February 2015

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?

$\displaystyle\frac{{1}}{{2}}+\frac{{1}}{{5}}+\frac{{1}}{{10}}+\frac{{1}}{{17}}+\frac{{1}}{{26}}+\ldots$ 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?