Properties of the convergence of series

 The objectives of this post are:

1) Determine the properties of the convergence of series

2) Do some exercises on the convergence and divergence of series

Properties of the convergence of series

Let's consider two series Σaₙ and Σbₙ. If they are convergent, then

1. Σcaₙ is convergent where c is any number and  Σcaₙ = cΣaₙ

2. Σaₙ ∓ Σbₙ are also convergent and Σaₙ ∓ Σbₙ = 𝚺(aₙ ∓ bₙ.)

Let's now do some exercises on the convergence and divergence of series

 Ex 1. Determine if the following series is convergent and determine its value.

Solution

Let's determine  the 5 first partial sums:

S₁ =  1

S₂ = S₁ + S₂ = 1 + 2 = 3

S₃ = S₁ + S₂ + S₃ = 3 + 3 = 6

S₄ = S₁ + S₂ + S₃  + S₄ = 6 + 4 = 10

S₅ = S₁ + S₂ + S₃  + S₄ + S₅ = 10 + 5 = 15

Let's find a pattern from the above partial sums. Let's re-write each partial sum:

S₁ =  1 = 1(1 + 1)/2 

S₂ =  3 = 2(2 + 1)/2

S₃ = 6 = 3(3 + 1)/2

S₄ = 10 = 4(4 + 1)/2 

S₅ = 15 = 5(5 + 1)/2

The pattern from the partial sum is n(n + 1)/2. The formula for the general term of the sequence of the partial sums is then n(n + 1)/2

The limit of the sequence of the partial sum is the limit of the sequence

The limit of this sequence is the limit of n(n + 1)/2 when n approaches infinity. By substituting n by infinity the limit is infinity. The limit of the sequence of the partial sums is infinity. This is also the limit of the infinite series. The series diverges. 

Ex 2. Same question with the sequence 

Let's play the same little game of finding the first partial sums:

S₁ = (-1)ⁱ = -1

S₂ =  S₁ + S₂ = -1 + (-1)² = -1 + 1 = 0

S₃ =  S₁ +S₂ + S₃ = 0 + (-1)³ = -1

S₄ =  S₁ +S₂ + S₃ + S₄  = -1 +  (-1)⁴ = -1 + 1 = 0

S₅ = S₁ +S₂ + S₃ + S₄ +S₅ = 0 +  (-1)⁵ = 0 -1 = -1

The sequence of the partial sums can be written as:

Sₖ = { 0, -1, 0, -1, -1, 0........}

When n approaches to infinity this sequence doesn't approach to any particular number since it repeats indefinitely the same pattern 0, -1, 0, -1. The sequence doesn't have a limit. Therefore it diverges and the infinite series diverges also.

Practice

Using the sequence of the partial sums, find the limit of the series

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