c Write two finite series and verify whether they satisfy the statement.
D
d Write two finite series and verify whether they satisfy the statement for an arbitrary value of c.
A
a True, see solution.
B
b True, see solution.
C
c False, see solution.
D
d False, see solution.
Practice makes perfect
a We want to determine whether the given statement is true.
∑_(i=1)^nc a_i ? = c∑_(i=1)^n a_i
The left hand-side of the equation is the sum of the first n terms in a sequence where each term is multiplied by a constant c.
∑_(i=1)^nc a_i = ca_1+ ca_2+ ... + ca_n
Let's factor out c.
∑_(i=1)^nc a_i = c(a_1+a_2+ ... + a_n)
Now, the right hand-side of this equation is c times the sum of the first n terms of the sequence.
∑_(i=1)^nc a_i & = c (a_1+a_2+ ... + a_n_(⇓))
∑_(i=1)^nc a_i & = c ∑_(i=1)^n a_i ✓
Therefore, the given statement is true.
b Again, we want to determine whether the given statement is true.
∑_(i=1)^n (a_i + b_i) ? = ∑_(i=1)^n a_i + ∑_(i=1)^n b_iLet's expand the series on left hand-side and rewrite it using the Commutative Property of Addition.
We see that the series can be written as the sum of the first n terms of two sequences, a_n and b_n.
As we can see, the given statement is true.
c We are asked to determine if the following statement is true.
∑_(i=1)^n a_i b_i ? = ∑_(i=1)^n a_i ∑_(i=1)^n b_i
The series on the left-hand side shows the products of the terms with the same position in their sequences. However, the expression on the right-hand side is the product of two series. Therefore, it appears that they are not equal. Let's consider two finite sequences.
l Sequence (I): 1,2,3 Sequence (II): 1,2,3 ⇒ l a_n=n b_n=n
We will first calculate the sum of the products a_i b_i for i=1, 2, and 3.
∑_(i=1)^3 a_i b_i & = ∑_(i=1)^3 i* i
& = 1* 1+ 2* 2+ 3* 3
& = 14
Now we will calculate the product of the series.
∑_(i=1)^3 a_i ∑_(i=1)^3 b_i & = (∑_(i=1)^3 i ) ( ∑_(i=1)^3 i )
& = ( 1+ 2+ 3)( 1+ 2+ 3)
& = 36
The calculations give different results.
14 ≠ 36
⇓
∑_(i=1)^n a_i b_i ≠ ∑_(i=1)^n a_i ∑_(i=1)^n b_i
Therefore, the given statement is false. Note that there are infinitely many counterexamples. We have shown only one of them.
d Let's think about the expressions on both side of the given statement.
∑_(i=1)^n (a_i)^c ? =( ∑_(i=1)^n a_i )^c
The series on the left-hand side shows the sum of each term raised to the power of c. However, on the right-hand side we have a series raised to the power of c. Intuitively, we may think that they are not equal. Let's consider a finite sequence to check this.
1,2,3 ⇒ a_n=n
We will choose an arbitrary value for c. For simplicity, let c= 2. We will first calculate the sum of the squares of the above sequence.
∑_(i=1)^3 (a_i)^2 & = ∑_(i=1)^3 i^2
& = 1^2+ 2^2+ 3^2
& = 14
Now we will calculate the square of the sum of the terms.
(∑_(i=1)^3 a_i )^2 & = (∑_(i=1)^3 i )^2
& = ( 1+ 2+ 3)^2
& = 36
The calculations give different results.
14 ≠ 36
⇓
∑_(i=1)^n (a_i)^c ≠ ( ∑_(i=1)^n a_i )^c
Therefore, the given statement is false. There are infinitely many counterexamples to prove this statement is false, and this is just one of them.
