Se ri e s o f re a l n um b e rs - Babeș-Bolyai Universitymath.ubbcluj.ro/~ancagrad/studenti/English/l5.pdf · C l assi c al operat i ons suc h as mul t i pl i c at i on wi t h a

Series of real numbers Group 814

November 2019 Lecture no.5

● De�inition 1

Each order of two sequences ((x n ),(s n )) n≥1 with the property that S k = x k S k+1 = x k + x k+1

… S n =x k + x k+1 + … + x n

is called a series of real numbers .

It is denoted by x n ∑

n≥k

Terminology ❖ (x n ) is the generating sequence and ∀ n∈ ℕ x n is the general term of degree n ❖ ∀ n∈ ℕ (s n ) ∈ ℝ is called a partial sum of degree n ❖ (s n ) the sequence of the partial sums

● De�inition 2

Let (x n ) let x n be a series of real numbers. If there exists , ∑

n≥k∈ ℝlim

n→∞sn = l

then l is said to be THE SUM of the series and is denoted by (= )∑∞

n=kxn l

● De�inition 3

The series (x n ) is said to be convergent if . It is divergent ∑

n≥k∃) ∈ℝ( ∑

∞

n=kxn

otherwise.

Remark❕ A series ⅀ x n is divergent in two cases:

1. ∃ { 士 ∞} limn→∞

sn =

2. ∄ limn→∞

sn

Examples:

❏ Studying the series we get that =1 so this series has a ∑

n≥1

1n(n+1) ∃ ∑

∞

n=1

1n(n+1)

sum and this is convergent

❏ Studying the series we get that ∑

n≥1( )− 1 n

the series is divergent and it doesn’t have a sum . ∄ limn→∞

sn ⇒

Geometric series

● De�inition

Let q≠0, then q n-1 is called the geometric series . ∑

n≥1

For k=1 and the sequence x n =q n-1 ,∀ n∈ℕ the sequence of the partial sums is: ● s n =x 1 =q 0 =1 ● s 2 =x 1 +x 2 =q 0 +q 1 =1+q

... ● s n =x 1 +x 2 +...+x n =q 0 +q 1 +...+q n-1

which is equal to either , for q≠1 or n , for q=1. 1 q−1 q − n

Case 1: q=1 ⇒ s n = =+∞ ⇒the geometric series is divergent and it has limn→∞

limn→∞

n

the sum =∞ ∑∞

n=11n

Case 2: q≠1(≠0)⇒ is equal to: ∑∞

n=1qn 1−

● ∞ ,q≥1 ● , 0< <1 1

1 q− q∣ ∣ ● ∄, q≤-1

The geometric series has a sum ∀ q∈(-1,∞)/{0} and it’s convergent ⇔0< <1 q∣ ∣

Telescopic series

● De�inition

Let x n such that ∀ n≥k x n =a n -a n+1 where (a n ) n ⊆R ∑

n≥kk ≥

● s k =x k =a k -a k+1 ● s k+1 =x k +x k+1 =a k -a k+2

... ● s n =x k +x k+1 +...+x n = a k -a n+1

(a k -a n+1 )=a k - a n ⇒if ∃ a n ⇒ x n =a k - a limn→∞

sn = limn→∞

limn→∞

limn→∞

∑∞

n=klimn→∞

The series is covergent ⇔ the sequence is convergent. A different starting index alters the sum , but keeps the nature ( convergent

remains convergent, divergent remains divergent) Classical operations such as multiplication with a scalar, addition, etc... can

be performed on series, but you have to pay attention to the non determined cases .

Let x n and y n be two series of real numbers and a,b from R. Then: ∑

n>k∑∞

n>k

(ax n +by n )=a x n +b y n ∑∝

n=k∑∞

n=k∑∞

n=k

Remark❕ Let x n be a series which has a sum. Then

x n =x k +x k+1 +...+x k+p-1 + x n ∑∞

n=k∑∞

n=k+p

Realizat de: Spatar Bogdan, So�ian Razvan, Timerman Alina, Solcan Cezar