Question

Calculate a geometric series

1.For v greater than 0 and less than 1, what is the Sum(v^i) for i = 1 to 100?

2.For v greater than 0 and less than 1, what is the Sum(v^i) for i = 1 to infinity?

3.Let v = 1/(1+r). State the answer to question 2 in terms of r.

Answer 1

In General we write a Geometric series like this:

{a, av, av², av³, ... }

• a is the first term, and
• v is the factor between the terms (called the "common ratio") and v ≠ 0.

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Formula for the n-th term can be defined as:

a_n = a_n-1⋅v
a_n = a⋅v^n-1

Now,

a) For 0<v<1, vⁱ for i = 1 to 100.

The geometric series :

v, v², v³, v⁴,.........................,v⁹⁹, v¹⁰⁰.

The sum of series is :

S₁₀₀ = v + v² + v³ +......................+ v⁹⁹ + v¹⁰⁰ --- eqⁿ 1

v.S₁₀₀ =  v² + v³ + v⁴......................+ v¹⁰⁰ + v¹⁰¹   -- eqⁿ 2

Subtract eqⁿ 2 from  eqⁿ 1, we get

S₁₀₀ - v.S₁₀₀ = v - v¹⁰¹

S₁₀₀ (1-v) = v- v¹⁰¹

S₁₀₀ = . -- eqⁿ 3

b) For 0<v<1, vⁱ for i = 1 to infinity

The geometric series :

v, v², v³, v⁴,.........................

The sum of series is :

S_∞= v + v² + v³ + v⁴ +........... +v∞

For finite n, we have S_n =

when n tends to infinity, vⁿ = 0 for 0<v<1,

Hence,

S_∞=  . --  eqⁿ 4

c)  Put v = 1/ (1+r) in  eqⁿ 4 , we get

S_∞ =    .