Let denote the sum of the first terms of an arithmetic progression with positive…

MATHEMATICS · Subjective · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Let

S_A(n)

denote the sum of the first

n

terms of an arithmetic progression with positive first term

a

and common difference

d>0

, and let

S_G(n)

denote the sum of the first

n

terms of a geometric progression with the same first term

a

and common ratio

r>1

. (i) Using the standard expressions for

S_A(n)

and

S_G(n)

, write an expression for the ratio

\dfrac{S_A(n)}{S_G(n)}

. (ii) Prove that

\displaystyle\lim_{n\to\infty}\dfrac{S_A(n)}{S_G(n)}=0

. (iii) Give a brief interpretation of the above result in the context of the relative growth of the two series.