Calculus II: Series definitions

1. Definitions

A series is a sequence of the sum of the terms of a sequence.

If {a_n} is a sequence, the associated series is {s_n}.

Sequence: a₁, a₂, ... , a_n, ...
Series: a₁ + a₂ + a_n + ...

The general term s_n of a series {s_n} is the partial sum of the terms of the associated sequence. It is equal to Σ a_k, where k goes from 1 to n.

s_n = Σ a_k
k: 1 → n

The series {S_n} is
Σ a_k
k from 1 to ∞

2. Convergence of a series

For any series of general term s_n, when:

lim sn	= L   (L∊ℝ)
n → ∞

the series {s_n} is convergent. If the limit L does not exist, or infinite, the series is divergent.

3. Examples

3.1. Example 1

Let's consider the sequence {1/n(n + 1)}:

The first terms of this sequence are:

a₁ = 1/2
a₂ = 1/6
a₃ = 1/12
a₄ = 1/20
...

Therefore, the first terms of the associated series {s_n} are:

s₁ = a₁ = 1/2
s₂ = a₁ + a₂ = 1/2 + 1/6 = 2/3
s₃ = a₁ + a₂ + a₃ = 1/2 = 1/6 + 1/12 = 3/4
...
s_n = a₁ + a₂ + a₃ + ... + a_n =
1/2 + 1/6 + 1/12 + ... + 1/n (n + 1).

s_n is called the partial sum of the first terms of the series.

The complete series is written as:

Σ a_n = a₁ + a₂ + a₃ + ... + a_n + ...
n from 1 to ∞

So, when we add the terms of a sequence, we obtain a infinite series (or simply a series).

If {a_n } is a sequence of real numbers, the series associated to this sequence is represented by the expression:

Σ a_n = a₁ + a₂ + a₃ + ... + a_n + ...
n = 1 → ∞

A series is the sum of an infinite number of numbers:
a₁ + a₂ + a₃ + ... + a_n + ...
which can have a finite sum. This finite sum is called the limit of the series, and the series is called convergent.

Now, let's find the general term or our series:

{s_n} = 1/2 + 1/6 + 1/12 + ... + 1/n (n + 1) + ...

The difficult task in series in to find its general term. Sometimes, it is evident, generally, we proceed by trial and error method. (we will see that we can use integrals as well).

For our series, it is clear that the general term is n/(n + 1).

Therefore

s_n = n/(n + 1).

S = lim sn	= lim n/(n + 1) = 1
n → ∞	n → ∞

Hence

The series {n/(n + 1)} is convergent.

3.1. Example 2

Let's consider a rod of length equal to 1. If one divides this rod in two parts (1/2, 1/2) and divides the second half in two parts (1/4,1/4), and so on, we get the following sequence when taking a half each time:

1/2, 1/4, 1/8, 1/16, ...

This sequence has the general term a_n = (1/2)ⁿ

If we add all the pieces of the rod, we will obtain the entire rod. That is if we add 1/2, 1/4, 1/8, 1/16, ... will we obtain 1.

We say that the limit of the series is 1, and we can write:

1/2 + 1/4 + 1/8 + 1/16 + ... + (1/2)ⁿ + ... = 1
or
s_n = 1/2 + 1/4 + 1/8 + 1/16 + ... + (1/2)ⁿ =
Σ (1/2)^k
k = 1 → n

{s_n} is the sequence of the partial sums associated to the sequence {a_n}

We can remark that the general term of the series sⁿ = 1 - a_n = 1 - (1/2)ⁿ

So

lim sn	= 1
n → ∞

The sires {1 - (1/2)ⁿ} is convergent.