Question

Find the sum of the following series to infinity:

(i) 1 – 1/3 + 1/3² – 1/3³ + 1/3⁴ + … ∞

(ii) 8 + 4√2 + 4 + …. ∞

(iii) 2/5 + 3/5² + 2/5³ + 3/5⁴ + …. ∞

(iv) 10 – 9 + 8.1 – 7.29 + …. ∞

Answer 1

(i) 1 – 1/3 + 1/3² – 1/3³ + 1/3⁴ + … ∞

Given:

S_∞ = 1 – 1/3 + 1/3² – 1/3³ + 1/3⁴ + … ∞

Where, a = 1, r = -1/3

By using the formula,

S_∞ = a/(1 – r)

= 1 / (1 – (-1/3))

= 1/ (1 + 1/3)

= 1/ ((3 + 1)/3)

= 1/ (4/3)

= 3/4

(ii) 8 + 4√2 + 4 + …. ∞

Given:

S_∞ = 8 + 4√2 + 4 + …. ∞

Where, a = 8, r = 4/4√2 = 1/√2 

By using the formula,

S_∞ = a/(1 – r)

= 8 / (1 – (1/√2))

= 8 / ((√2 – 1)/√2) 

= 8√2 /(√2 – 1)

Multiply and divide with √2 + 1 we get,

= 8√2 /(√2 – 1) × (√2 + 1)/( √2 + 1)

= 8 (2 + √2)/(2 - 1)

= 8 (2 + √2)

(iii) 2/5 + 3/5² + 2/5³ + 3/5⁴ + …. ∞

The given terms can be written as,

(2/5 + 2/5³ + …) + (3/5² + 3/5⁴ + …)

(a = 2/5, r = 1/25) and (a = 3/25, r = 1/25)

By using the formula,

S_∞ = a/(1 – r)

(iv) 10 – 9 + 8.1 – 7.29 + …. ∞

Given:

S_∞ = 8 + 4√2 + 4 + …. ∞

Where, a = 10, r = -9/10

By using the formula,

S_∞ = a/(1 – r)

= 10 / (1 – (-9/10))

= 10 / (1 + 9/10)

= 10 / ((10 + 9)/10)

= 10 / (19/10)

= 100/19

= 5.263