Answer:
The sigma notation of the series is 8∑n=2 [1/2(1/2)^n-2]
Step-by-step explanation:
* Lets revise the meaning of sigma notation
- A series can be represented in a compact form, called summation or
sigma notation.
- The Greek capital letter, ∑ , is used to represent the sum.
# Ex:
- The series 4 + 8 + 12 + 16 + 20 + 24 can be expressed 6Σ(n=1) 4n
- The expression is read as the sum of 4n as n goes from 1 to 6
- The coefficient of n is 4 because the common difference is 4
- The variable n is called the index of summation.
# Look to the attached figure to more understand
* Now lets solve the problem
∵ The series is 1/2 + 1/4 + 1/8 + ........... + 1/128 and n = 2
- There is a constant ratio between each to consecutive terms
∴ The series is geometric
∴ a1 = a , a2 = ar , a3 = ar² , a4 = ar³ , ..........
∵ an = a(r)^n-1, where a is the first term, r is the common ratio and n
is the position of the number in the series and the first n is 1
∵ 1/4 ÷ 1/2 = 12 , 1/8 ÷ 1/4 = 1/2
∴ The constant ratio is 1/2
∵ The first term is 1/2
∵ The last term is 1/128
∵ 1/128 = 1/2^7
∴ 1/128 = (1/2)^7
- There are seven terms in the sequence
∵ The n of the first term is 2
∴ The n of the last term = 7 + 1 = 8
* Now lets write the sigma notation
∴ 8∑n=2 [1/2(1/2)^n-2] ⇒ put them like the attached figure
- To generate the terms of the series given in sigma notation above,
replace n by 2 , 3 , 4 , 5 , 6 , 7 , 8
