Find the sum of the series if |7x|<1.?

Question

Find the sum of the series if |7x| < 1.
1 - (7x) + (7x)^2 - (7x)^3 + (7x)^4 +....
Find the sum of the series if |7y| < 1.
1 - (7y)^2 + (7y)^4 - (7y)^6 + .....

Answer 1

You know the geometric series

1- r + r^2 - r^3 + ...... = 1/(1+r) if |r| <1

So in each problem, what quantity (less than one) can you set equal to r in order to use this formula?

Answer 2

q1
if 0 <= 7x < 1,
sum = 1/[1-(7x)^2] - 7x/[1-(7x)^2]
= (1-7x)/[1-(7x)^2]
= 1/(1+7x)

if -1 < 7x <= 0,
sum
= 1/(1-7x)

so answer is
sum = 1/(1+7x)

q2
sum = 1/[1+(7y)^2]
= 1/(1+49y^2)