Counterexample: how about
The harmonic series:
is divergent, i.e. sums to .
This geometric series:
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 from the harmonic series, it would remain divergent. is still 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?
Is Calculus the right forum for questions on series and sequences?
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.