4 - Number series Flashcards
Convergence. Geometric, harmonic, telescoping series and their convergence or divergence. Series of positive numbers, comparison test.
What are number series?
Similar concept to a sequence. Informally, a series is an infinite sum like this:
a0 + a1 + a2 + a3 + … or Σ a_k
We call Sn the n-th partial sum of Σ a_k and (Sn) the sequence of partial sums.
What is an example of a number series?
The Geometric series is a number series.
q ϵ C, Σ q^n, a_n = q^n, n>=0
Sn = 1 + q + q^2 + … + q^n = Σ q^k
Where q is called the quotient of the geometric series.
How do you form a generic geometric series expression?
Sn = 1 + q + … + q^n
q*Sn = q + q^2 + … + q^(n+1)
Sn - qSn = 1 - q^(n+1)
Sn(1-q) = 1 - q^(n+1)
Sn = (1 - q^(n+1)) / (1-q)
When do geometric series converge?
Geometric series converges if and only if the absolute value of the quotient is less than 1.
What is the Leibnitz Criterion?
The alternating series test is the method used to prove that an alternating series with terms that decrease in absolute value is a convergent series. The test was used by Gottfried Leibniz and is sometimes known as Leibniz’s test, Leibniz’s rule, or the Leibniz criterion.
What is the harmonic series?
Sn = 1 + 1/2 + 1/3 + 1/4 + … + 1/n = Σ 1/n
Does the Cauchy Criterion work for series as for sequences?
Yes. A series Σ a_n is convergent if and only if for each epsilon greater than 0, there is a natural number, n0, such that:
^mΣ_(k=n+) a^k | < ε
Can the Leibnitz test determine convergence on its own?
No, it helps to determine convergence but other tests needed as well
What are non-negative series?
The sequence of partial sums is increasing if a_k >= 0 for all natural numbers k, because then:
Sn = Σ a_k <= a_(n+1) + Σ a_n = S(n+1)
for all natural numbers, n. We call such a series a non-negative series.
Are non-negative series convergent?
A series with non-negative terms is convergent if and only if its sequence of partial sums is bounded.
What is a telescoping series?
Telescoping series is a series that can be rewritten so that most (if not all) of the terms are cancelled by a preceding or following term. Generally, the first and last terms are left
What is the Comparison Test?
Suppose that we have two series ∑an and ∑bn with an, bn ≥ 0 for all n and an ≤ bn for all n. Then,
- If ∑bn is convergent then so is ∑an
- If ∑an is divergent then so is ∑bn.
The basis of the direct comparison test is that if every term in one series is less than the corresponding term in some convergent series, it must converge as well.
In other words, we have two series of positive terms and the terms of one of the series are always larger than the terms of the other series. Then if the larger series is convergent the smaller series must also be convergent. Likewise, if the smaller series is divergent then the larger series must also be divergent. Note as well that in order to apply this test we need both series to start at the same place.
The comparison test compares two similar series and uses the easy one to find the hard one.
What is the limit comparison test?
Suppose that we have two series ∑an and ∑bn with an ≥ 0, bn > 0 for all n. Define,
c = lim_(n→∞) an / bn
If c is positive (i.e. c > 0) and is finite (i.e. c < ∞) then either both series converge or both series diverge.
In other words (an) is convergent if and only if (bn) is convergent. Vice versa and also true for divergent series.
What is the p-series?
A p-series is any one of a family of infinite series whose terms are reciprocal powers of natural numbers. The parameter p∈{R} specifies the power, which defines the series.
Any p-series can be expressed in summation notation using the following general formula:
∞∑n=1/n^p
What is the p-series test for convergence?
Whether or not a p-series converges to a specific value can be determined using the very simple p-series test for convergence. The test states that the series;
∞∑n=1/n^p
will converge if p>1, otherwise it will diverge.
