Math Help - series questions, not a problem.

1. series questions, not a problem.

Are the geometric, harmonic, and collapsing series all the same?

Or can i only test the geometric series by this formula: a/(1-r)

is it that the harmonic series ALWAYS diverges.
Is this because it continues on infinitly?

and the collapsing series, In order to determine if it converges, must i first put the equation in a two fractions form, through partial fraction substitution for example?

If i need to show specific problem examples i can, any further explanation to my questions can help. Just using basic calculus 3 skills here. thankyou.

2. Originally Posted by rcmango
Are the geometric, harmonic, and collapsing series all the same?
.
I do not know what you mean. They are all three different series.

Or can i only test the geometric series by this formula: a/(1-r)

is it that the harmonic series ALWAYS diverges.
Is this because it continues on infinitly?

and the collapsing series, In order to determine if it converges, must i first put the equation in a two fractions form, through partial fraction substitution for example?

If i need to show specific problem examples i can, any further explanation to my questions can help. Just using basic calculus 3 skills here. thankyou.
In other words, how do i know the difference between a harmonic series and a geometric series?

4. Originally Posted by rcmango
Are the geometric, harmonic, and collapsing series all the same?

Or can i only test the geometric series by this formula: a/(1-r)

is it that the harmonic series ALWAYS diverges.
Is this because it continues on infinitly?

and the collapsing series, In order to determine if it converges, must i first put the equation in a two fractions form, through partial fraction substitution for example?

If i need to show specific problem examples i can, any further explanation to my questions can help. Just using basic calculus 3 skills here. thankyou.
Harmonic Series:

$\sum_{k=1}^{\infty}\frac{1}{k}$

diverges. It diverges because the terms do not decrease fast enough for
the sum to converge. The terms can be grouped into ever larger groups each
of which has a sum greater than (or equal) to 1/2, and because there are still
an infinite number of such groups the sum diverges (for more details see either
the wikipedia or mathworld pages on the harmonic series)

The n-th parial sum of the harmonic series:

$H_n=\sum_{k=1}^{n}\frac{1}{k}$

is the n-th harmonic number

RonL

5. Originally Posted by rcmango
Are the geometric, harmonic, and collapsing series all the same?
Collapsing series? Is this the same as a telescoping series?

A telescoping series is one in which the general term $a_n$ may
be written as the difference of consecutive terms of a sequence, that is:

$a_n=b_n-b_{n+1}$.

Then:

$\sum_1^N a_n = \sum_{n=1}^N (b_n-b_{n+1})$

but now we have cancellation of all the $b's$ except the first and last, so:

$\sum_1^N a_n = \sum_{n=1}^N (b_n-b_{n+1})=b_1-b_{N+1}$

RonL