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