Infinite series - convergent vs divergent

ScottO · #1 $Old$ September 19th, 2007, 11:11 AM

The harmonic series:

$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ...$

is divergent, i.e. sums to $\infty$ .

This geometric series:

$\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$

is convergent to 2.

In general, is there a way to know if removing all the elements of a convergent series, from a divergent series, if the result will be divergent or convergent?

It makes sense to me, that if I were to remove $\frac{1}{1}$ from the harmonic series, it would remain divergent. $\infty - 1$ is still $\infty$ after all. But how much can I take away from something infinite, and have it remain so? Thinking the other way, how much do I have to remove to get it to converge?

-Scott

P.S.
Is Calculus the right forum for questions on series and sequences?

ThePerfectHacker · #2 $Old$ September 19th, 2007, 11:35 AM

Counterexample: how about $1+1+1+...$

CaptainBlack · #3 $Old$ September 19th, 2007, 01:49 PM

Quote:

Originally Posted by ScottO $View Post$

The harmonic series:

$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ...$

is divergent, i.e. sums to $\infty$ .

This geometric series:

$\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$

is convergent to 2.

In general, is there a way to know if removing all the elements of a convergent series, from a divergent series, if the result will be divergent or convergent?

It makes sense to me, that if I were to remove $\frac{1}{1}$ from the harmonic series, it would remain divergent. $\infty - 1$ is still $\infty$ after all. But how much can I take away from something infinite, and have it remain so? Thinking the other way, how much do I have to remove to get it to converge?

-Scott

P.S.
Is Calculus the right forum for questions on series and sequences?

Removing a convergent subseries from a divergent series will always leave it
divergent. Consider the sequence of partial sums:

S_n = C_n + D_n

where C_n is the partial sum of the convergent series terms which occur
before the n-th term of the original series, and D_n is the partial sum of the
remaininng terms up to the n-th term.

As n becomes large S_n goes to infinity while C_n goes to a finite limit, hence D_n goes to infinity.

RonL

ScottO · #4 $Old$ September 20th, 2007, 09:28 AM

Quote:

Originally Posted by CaptainBlack $View Post$

Removing a convergent subseries from a divergent series will always leave it
divergent. Consider the sequence of partial sums:

S_n = C_n + D_n

where C_n is the partial sum of the convergent series terms which occur
before the n-th term of the original series, and D_n is the partial sum of the
remaininng terms up to the n-th term.

As n becomes large S_n goes to infinity while C_n goes to a finite limit, hence D_n goes to infinity.

RonL

What if only divergent series were considered? Are there different degrees (sizes?) of divergent series such that removing one from another may or may not lead to a convergent result?

-Scott

CaptainBlack · #5 $Old$ September 20th, 2007, 09:59 AM

Quote:

Originally Posted by ScottO $View Post$

What if only divergent series were considered? Are there different degrees (sizes?) of divergent series such that removing one from another may or may not lead to a convergent result?

-Scott

Any sequence such that the absolute values of the terms is eventualy decreasing with limit 0 contains a subsequence whos sum is a convergent series.

RonL