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.
how about the other questions please.
In other words, how do i know the difference between a harmonic series and a geometric 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.
Harmonic Series:
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:
is the n-th harmonic number
RonL
Collapsing series? Is this the same as a telescoping series?
A telescoping series is one in which the general term may
be written as the difference of consecutive terms of a sequence, that is:
.
Then:
but now we have cancellation of all the except the first and last, so:
RonL