Testing an Infinite Sum of a Series Made up of a Difference for Convergence

Hi,

I am asked to determine if the infinite sum of [1/5n - 1/(5n+3)] is convergent.

I tested each term individually and found that they both diverge, which means that I cannot say whether their difference converges or diverges.

Can anyone tell me what I should do when I encounter a difference or sum of series and they both diverge individually?

I found that the series above is greater than [1/5n^2 - 1(5n+3)] and that the first term in this converges while the second diverges still. So wouldn't that mean that their difference diverges, and that the original series diverges by the Comparison Test?

WolframAlpha says it converges ???

infinite sum (1/(5n)-1/(5n+3)) - Wolfram|Alpha

Any help? What should I do?

Thanks

Re: Testing an Infinite Sum of a Series Made up of a Difference for Convergence

Hint: try to put the terms together into one fraction. Then you can apply this general rule:

**If you are dealing with the ratio of two polynomials, the series will converge if and only if the highest power of the denominator exceeds the highest power of the numerator by **__more__ than 1.

For example,

$\displaystyle \sum_{n=1}^{\infty} \frac{n^2+10000000n}{n^4}$ will converge

On the other hand,

$\displaystyle \sum_{n=1}^{\infty} \frac{n^3+10n}{100000n^4}$

Will diverge.

See if you can apply the rule in this situation.

Re: Testing an Infinite Sum of a Series Made up of a Difference for Convergence

Quote:

Originally Posted by

**SworD** Hint: try to put the terms together into one fraction. Then you can apply this general rule:

**If you are dealing with the ratio of two polynomials, the series will converge if and only if the highest power of the denominator exceeds the highest power of the numerator by **__more__ than 1.

For example,

$\displaystyle \sum_{n=1}^{\infty} \frac{n^2+10000000n}{n^4}$ will converge

On the other hand,

$\displaystyle \sum_{n=1}^{\infty} \frac{n^3+10n}{100000n^4}$

Will diverge.

See if you can apply the rule in this situation.

Okay thanks :) I will try that. But can you say that the infinite sum of a difference is divergent if one of the individual sums is convergent and the other is divergent?

Re: Testing an Infinite Sum of a Series Made up of a Difference for Convergence

Quote:

Originally Posted by **Coop;780093**

I am asked to determine if the infinite sum of [1/5n - 1/(5n+3)

is convergent.

This is really just a simple **comparison test**.

$\displaystyle \sum\limits_{n = 1}^\infty {\left( {\frac{1}{{5n}} - \frac{1}{{5n + 3}}} \right)} = \sum\limits_{n = 1}^\infty {\left( {\frac{3}{{25{n^2} + 15n}}} \right)} \leqslant \frac{3}{{25}}\sum\limits_{n = 1}^\infty {\left( {\frac{1}{{{n^2}}}} \right)} $

Can you see the comparison?

Re: Testing an Infinite Sum of a Series Made up of a Difference for Convergence

Quote:

Originally Posted by

**Plato** This is really just a simple **comparison test**.

$\displaystyle \sum\limits_{n = 1}^\infty {\left( {\frac{1}{{5n}} - \frac{1}{{5n + 3}}} \right)} = \sum\limits_{n = 1}^\infty {\left( {\frac{3}{{25{n^2} + 15n}}} \right)} \leqslant \frac{3}{{25}}\sum\limits_{n = 1}^\infty {\left( {\frac{1}{{{n^2}}}} \right)} $

Can you see the comparison?

Right thanks, I think that's where SworD was going. I see the comparison but I am having a little trouble seeing your first equality. :/

P.S. Is my statement above your post true? About the infinite sum of a difference converging.

Re: Testing an Infinite Sum of a Series Made up of a Difference for Convergence

Quote:

Originally Posted by

**Coop** I see the comparison but I am having a little trouble seeing your first equality.

Are you saying that you have trouble adding two fractions?

Re: Testing an Infinite Sum of a Series Made up of a Difference for Convergence

Nvm I figured it out, thanks for the help SworD and Plato.

Re: Testing an Infinite Sum of a Series Made up of a Difference for Convergence

Quote:

Originally Posted by

**Coop** But can you say that the infinite sum of a difference is divergent if one of the individual sums is convergent and the other is divergent?

This is true.

Quote:

Originally Posted by

**Coop** I found that the series above is greater than [1/5n^2 - 1(5n+3)] and that the first term in this converges while the second diverges still. So wouldn't that mean that their difference diverges, and that the original series diverges by the Comparison Test?

It is also true that $\displaystyle \sum_{n=1}^\infty(1/(5n^2) - 1/(5n+3))$ diverges and that $\displaystyle 1/(5n) - 1/(5n+3)\ge1/(5n^2) - 1/(5n+3)$. However, it is not true that $\displaystyle |1/(5n) - 1(5n+3)|\ge|1/(5n^2) - 1/(5n+3)|$. If the latter inequality were true, then the original series would diverge by the comparison test.

Re: Testing an Infinite Sum of a Series Made up of a Difference for Convergence

Quote:

Originally Posted by

**emakarov** This is true.

It is also true that $\displaystyle \sum_{n=1}^\infty(1/(5n^2) - 1/(5n+3))$ diverges and that $\displaystyle 1/(5n) - 1/(5n+3)\ge1/(5n^2) - 1/(5n+3)$. However, it is not true that $\displaystyle |1/(5n) - 1(5n+3)|\ge|1/(5n^2) - 1/(5n+3)|$. If the latter inequality were true, then the original series would diverge by the comparison test.

Thank you :)