Note 8.2 - Introduction to Series
1 Introduction
A series is a sequence defined from a sequence. It is the discrete version of integration that bears many properties in common.
2 Definition and Examples
Definition 2.1. Given a sequence {ak}, we define a series {sn} by
sn=
n
X
k=k0
ak.
Each snis called the nthpartial sum. We say that the series converges if the partial sums converge and denote the limit by
It is usually not easy to determine whether a series converges. However, there are instances that series obviously diverge:
Theorem 2.2 (The kth Term Test). In the notations above, if the series di- verges, then ak→ 0
In the other words, if ak 9 0, then the series diverges.
Even if the series converges, very often we have no idea what value it con- verges to.
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3 Elementary Examples
We are probably familiar with this series. Given a, r ∈ R, let ak = ark. We have s0= a and
sn= a(1 − rn) 1 − r for n ≥ 1.
It is then not hard to tell that sn converges if and only if |r| < 1 and s = lim
n→∞sn= a 1 − r,
The other series with easy computable limit is called the telescoping series:
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4 Convergence Tests
As mentioned before, we are often more concerned on whether the series con- verge over the actual limiting value. Some of them are easier to apply than the others, but often come with the tradeoff of applicability or conclusiveness. Let’s skim through them below.
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5 Power Series
Power series is a polynomial of infinite degree. It is formally written as
P (x) =
∞
X
k=0
akxk.
Evidently, the convergence behavior of P depends on the value of x. It certainly converges for x = 0, but can diverge for other x’s:
It is a theorem that a power series always converge on an interval (−r, r), called convergence interval, for r ∈ (0, ∞]. This interval is usually determined by ratio or root test:
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On the next and final note, we will study a power series of particular importance.
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