Proof: Let be a real number such that , and
for all
let

Then by theorem 2.22 we have

and hence

Hence by the th power theorem

Let . Then, by the translation rule

Thus by the sum rule and product rule,

Now

Hence we have shown that

for all . (This sort of argument,

``The clearest early account of the summation of geometric series''[6, page 136] was given by Grégoire de Saint-Vincent in 1647. Grégoire's argument is roughly as follows:

On the line mark off points , , etc. such that

On a different line through mark off points , , , etc. such that

Then

Now I use the fact that

(see exercise 6.78), to say that

It follows that the triangles , , , etc. are all mutually similar, so the lines , , etc. are all parallel. Draw a line through parallel to and intersecting at . I claim that

It is clear that any finite sum is smaller that , and by taking enough terms in the sequence we can make arbitrarily small. Then is arbitrarily small, i.e. the finite sums can be made as close to as we please. By similar triangles,

so, equation (6.77) says

- a)
- Find .
- b)
- Find .
- c)
- For each in
let

(in part (b) you calculated ). Find a formula for , and then find . - d)
- Show that

(Thus it is not necessarily true that the limit of an infinite sum is the infinite sum of the limits. The left side of (6.80) was calculated in part c. The right side is , where depends on , but not on .)