Does the infinite geometric series diverge or converge? Explain.

Question

1/4 + 1/20 + 1/100 + 1/500

a. it converges; it has a sum.

b. it converges; it does not have a sum.

c. it diverges; it has a sum.

d. it diverges; it does not have a sum.

Answer 1

The answer is (c).

1) Find the ratio between first and second term. In this case it's 1/5

2) Ratio, r = 1/5. If |r|<1. Then series converges.

If not then series diverges and has no sum.

3) In this problem |r|<1, so it converges but we need to find out if it has a sum or not.

4) Formula for finding sum: a/1-r. Here a is the first term which is 1/4

5) Do the math (1/4)/(1-(1/5))= 5/16.

So the answer: (c) converges and has a sum

Answer 2

The pattern given to us is an infinite series with function ∑1/(4(5^(k-1))). You can see this by factoring out a 4 from each term's denominator, which then leaves us with just expressions of 5^(k-1) (with n starting at k=1). We can rewrite this as ∑(1/4)(1/5^(k-1)). This will come in handy rewriting it this way.

We know that it's an infinite geometric series. Any infinite geometric series where r, the rate at which we are multiplying each term, is less than 1, there is a sum. That's all the question asks you for -- does it converge? In this case, yes, it converges, and it has a sum.

BONUS:

To find the sum, use the formula S = a_1/(1-r), where a_1 is the first term of the series (k=1) and r Is the multiplier.

S = a_1/(1-r)

S = (1/4)/(1-(1/5))

= 5/16