Infinite Sequences, Limits and Series

Comp.145  January 1996  Dr. A.J. Mestel

(1) An infinite sequence is an ordered list of numbers or terms, which we write as {u_{n}} for n = 0, 1, 2 … The numbers u_{n} may be complex or real.

(2) The sequence {u_{n}} has a limit U, if, for all n sufficiently large, |u_{n} − U| is arbitrarily small. In that case we say the sequence converges, and write

u_{n} → U  as n → ∞ ,  or lim_{n→∞} u_{n} = U .

If the sequence does not tend to a limit, we say it diverges. For example, if u_{n} = 1/n, then u_{n} → 0 as n → ∞. The sequence {r^{n}}, where r is given, has the limit

lim_{n→∞} r^{n} = \{ \begin{matrix}0  if & |r| < 1 \\ ∞  if & |r| > 1 \end{matrix}

If |r| = 1, the sequence does not converge unless r = 1, when it has the limit 1.

(3) Limits behave well under addition and multiplication, so that if a_{n} → A, and b_{n} → B, then (a_{n} + b_{n}) → (A + B) and (a_{n}b_{n}) → AB.

(4) Polynomials in n are dominated as n → ∞ by their largest power of n. Thus

lim_{n→∞} (\frac{n^{p} + a_{1}n^{p−1} + … + a_{p}}{n^{q} + b_{1}n^{q−1} + … + b_{q}} ) = \{ \begin{matrix}0  if p & < q \\ 1  if p & = q \\ ∞  if p & > q \end{matrix}

Exponentials dominate polynomials. For example, lim_{n→∞} n^{100}(1.01)^{−n} = 0.

(5) The behaviour of any finite number of terms is irrelevant to whether or not a sequence converges.

(6) An infinite series is the sum of all the terms in an infinite sequence. We interpret

this infinite sum by defining the sequence of “partial sums,” s_{n} = ∑_{r=0}^{n} u_{r}. If the sequence

{s_{n}} tends to a finite limit S as n → ∞, we say that the infinite series converges to S, and write

∑_{n=0}^{∞} u_{n} = S .

Otherwise, we say that the series is divergent. The study of infinite series is a difficult topic, full of traps, especially when the terms may be either positive or negative.

(7) For a series ∑_{n=0}^{∞} u_{n} to converge, it is necessary that lim_{n→∞} u_{n} = 0, i.e. the terms

must tend to zero as n → ∞. Note that this condition is not sufficient to guarantee convergence.

(8) The comparison test. Suppose that a_{n} and b_{n} are real and a_{n} ⩾ b_{n} ⩾ 0. Then

(i) if ∑_{n=0}^{∞} a_{n} is convergent then ∑_{n=0}^{∞} b_{n} also converges.

(ii) if ∑_{n=0}^{∞} b_{n} is divergent, then so is ∑_{n=0}^{∞} a_{n}.

(9) The geometric series ∑_{n=0}^{∞} r^{n} converges if and only if |r| < 1. If r = 1 it increases

without limit, while if r = −1 it oscillates between 0 and 1.

(10) The harmonic series ∑_{n=1}^{∞} \frac{1}{n} diverges. More generally, the series ∑_{n=1}^{∞} \frac{1}{n^{s}}, where s is

real, converges iff s > 1. The behaviour of this series when s is complex is related to the most famous unsolved problem in mathematics, the Riemann hypothesis.

(11) The ratio test. Essentially using the comparison test with a geometric series we can show that if

lim_{n→∞} |\frac{a_{n+1}}{a_{n}} | = l , then the series ∑_{n=0}^{∞} a_{n} \{ \begin{matrix}converges if l & < 1 \\ diverges if l & > 1 \\ uncertain if l & = 1 \end{matrix}

This test is very useful, but sometimes the ratio does not tend to a limit.

(12) Power series: If {a_{n}} is a given sequence, we can define a function f(x)

f(x) = ∑_{n=0}^{∞} a_{n}x^{n} , provided the series converges.

We use the ratio test, defining u_{n} = a_{n}x^{n}, to infer that if

lim_{n→∞} |\frac{a_{n+1}}{a_{n}} | = \frac{1}{R} then lim_{n→∞} |\frac{u_{n+1}}{u_{n}} | = lim_{n→∞} |\frac{a_{n+1} x^{n+1}}{a_{n}x^{n}} | = \frac{|x|}{R}

so that the series converges if |x| < R, diverges if |x| > R, and may do either if |x| = R. This critical value of |x| is called the Radius of Convergence of the power series.

(13) If |x| < R, we can differentiate the power series for f(x) term by term m times, and then evaluate at x = 0 to find f^{(m)}(0) = a_{m}m!. This leads to the Maclaurin series:

f(x) = ∑_{n=0}^{∞} \frac{f^{(n)}(0)}{n!}x^{n} .

(14) A change of variable leads to the Taylor series for f(x + h):

f(x + h) = f(x) + hf′(x) + \frac{1}{2}h^{2}f′′(x) + … \frac{f^{(n)}(x)}{n!}h^{n} + O(h^{n+1}) .

This series is very useful when seeking approximations when h is small.