Fundamentals of Series: definition, convergence, divergence, index shift

The objectives of this post are:

1) Define a series

2) Define its limit

3) Define its convergence and divergence

4) Define index shift

5) Practice some exercises on index shift

Definition of a series

Let's consider the infinite sequence {a₁, a₂, a₃,........aₙ, ....}. It can also be written {aₙ} with n varying from 1 to infinite.

Let's write the sum of the infinite number of the terms of the sequence:

S =  a₁ + a₂ + a₃,........aₙ + ......

This sum is called an infinite series. It can also be written as:

We can also write the sum of the first term, the sum of the two first terms, the sum of the first three terms, the sum of the n terms of the sequence.

S₁ = a₁

S₂ = a₁ + a₂

S₃ = a₁ + a₂ + a₃

Each of these sums is called a partial sum.

Limit of the infinite series

The partial sums above form a sequence {sₙ} where n varies from 1 to infinity. The limit of this infinite sequence is the limit of sₙ when i varies from 1 to infinity.

 

The limit of the sequence of the partial sums {sₙ }is equal to the infinite series. The infinite series can be defined as the limit of the sequence of the partial sums of the sequence {aₙ}.

Convergence and Divergence

If the sequence of the partial sums {sₙ } converges meaning that its limit is finite, the infinite series converges. Otherwise the infinite series diverges.

Index shift

In the series

the letter i represent the index of the series. Here the series begins with i = 1 but we can start it at any value.

Example

Let's consider the following series 

,start the series at n = 0

Solution

In order to start the series at n = 0, we need to change the index. Let's call the new index i but we can't just replace n by i.

In order to change the index from 2 terms before, we have to write the new index as i = n-2 . For n = 2 the new lower term of the series is i = 0. For n = ∞, the value of the index is i = ∞-2 = ∞. The upper value doesn't change. From 

i = n-2 we have n = i + 2. Let's plug n in the series to perform the index shift. 

Let's substitute i by n we have:

In fact the 2 series are identical. Let's prove it by calculating the first few terms of each series:

The two series have the same terms. When we examine the two series, we can see that by decreasing the index by 2 the new terms increase by 2 . In the new series the index decreases from 2 to 0. The term n + 5 increases to n + 7 and the term 2ⁿ increases to 2²ⁿ⁺².

We can also deduct that when the index increases the terms of the series decreases. In performing the index shift, we just have to apply the rules above.

Example

  Solution 

If the index decreases the term of the series should increase. Since the index decreases by 1 we increase the term of the series by 1.

Practice