Infinitely decreasing geometric progression

Sum of an infinite geometric progression

Consider the sum of all terms

b_1+b_2+b_3+\cdots

. If

|q|<1

and the series converges, its sum is given by one formula:

S=b_1+b_1 q+b_1 q^2+\cdots=\dfrac{b_1}{1-q}

. Intuition: as

n\to\infty

q^n\to 0

, so in the

S_n

formula the limit is

\dfrac{b_1}{1-q}

Infinitely decreasing g.p. means

|q|<1

(school focus: decreasing modulus when

b_1>0,\ q>0

)

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