Pure 1.3 - Series

Question 1

Describe sigma notation.

Answer 1

The numbers below and above the Σ tell you which value of 𝑟 to begin at, and which value to end at. You go up by increments of 1 each time.
An example of its notation is Σ(ⁿᵣ₌₁)𝑟=1+2+3…+𝑛

Question 2

How do you find the sum of a series of constant terms?

Answer 2

For the constant term 𝑐, use the formula Σ(ⁿᵣ₌₁)𝑐=𝑐𝑛.

Question 3

How do you find the sum of the first 𝑛 natural numbers?

Answer 3

Use the formula Σ(ⁿᵣ₌₁)𝑟=½𝑛(𝑛+1).

Question 4

How do you find the sum of a series that doesn’t start at 𝑟=1?

Answer 4

Use Σ(ⁿᵣ₌ₖ)f(𝑟)=Σ(ⁿᵣ₌₁)f(𝑟)-Σ(ᵏ⁻¹ᵣ₌₁)f(𝑟).

Question 5

How are sums of more complicated series calculated?

Answer 5

Σ(ⁿᵣ₌₁)kf(𝑟)=kΣ(ⁿᵣ₌₁)f(𝑟)

Σ(ⁿᵣ₌₁)(f(𝑟)+g(𝑟))=Σ(ⁿᵣ₌₁)f(𝑟)+Σ(ⁿᵣ₌₁)g(𝑟)

Question 6

How is the sum of the series of squares of the first 𝑛 natural numbers calculated?

Answer 6

Use Σ(ⁿᵣ₌₁)𝑟²=⅙𝑛(𝑛+1)(2𝑛+1).

Question 7

How is the sum of the series of cubes of the first 𝑛 natural numbers calculated?

Answer 7

Use Σ(ⁿᵣ₌₁)𝑟³=¼𝑛²(𝑛+1)².