11.2

11.2 _Series

Series

What do we mean when we express a number as an

infinite decimal? For instance, what does it mean to write π = 3.14159 26535 89793 23846 26433 83279 50288 . . . The convention behind our decimal notation is that any

number can be written as an infinite sum. Here it means that

where the three dots (^{. . .}) indicate that the sum continues forever, and the more terms we add, the closer we get to

Series

In general, if we try to add the terms of an infinite sequence we get an expression of the form

a₁ + a₂ + a₃+ ^{. . .} + a_n + ^{. . .}

which is called an infinite series (or just a series) and is denoted, for short, by the symbol

Series

It would be impossible to find a finite sum for the series 1 + 2 + 3 + 4 + 5 + ^{. . .} + n + ^{. . .}

because if we start adding the terms we get the cumulative sums 1, 3, 6, 10, 15, 21, . . . and, after the nth term, we get n(n + 1)/2, which becomes very large as n increases.

However, if we start to add the terms of the series

we get

Series

The table shows that as we add more and more terms, these partial sums become closer and closer to 1.

Series

In fact, by adding sufficiently many terms of the series we can make the partial sums as close as we like to 1.

So it seems reasonable to say that the sum of this infinite series is 1 and to write

We use a similar idea to determine whether or not a general series (1) has a sum.

Series

We consider the partial sums s₁ = a₁

s₂ = a₁ + a₂

s₃ = a₁ + a₂ + a₃

s₄ = a₁ + a₂ + a₃ + a₄ and, in general,

s_n = a₁ + a₂ + a₃ + ^{. . .} + a_n =

These partial sums form a new sequence {s_n}, which may or may not have a limit.

Series

If lim_{n →} s_n = s exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series Σ a_n.

Series

Thus the sum of a series is the limit of the sequence of partial sums.

So when we write a_n = s, we mean that by adding

sufficiently many terms of the series we can get as close as we like to the number s.

Notice that

Example 2

An important example of an infinite series is the geometric series

a + ar + ar² + ar³ + ^{. . .} + ar^n–1 + ^{. . .} = a ≠ 0

Each term is obtained from the preceding one by multiplying it by the common ratio r.

If r = 1, then s_n = a + a + ^{. . .} + a = na → ± .

Since lim_{n →} s_n doesn’t exist, the geometric series diverges in this case.

Example 2

If r ≠ 1, we have

s_n = a + ar + ar² + ^{. . .} + ar^n-1

and rs_n = ar + ar² + ^{. . .} + ar^n-1 + arⁿ

Subtracting these equations, we get s_n – rs_n = a – arⁿ

cont’d

Example 2

If –1< r < 1, we know that as rⁿ → 0 as n → , so

Thus when | r | < 1 the geometric series is convergent and its sum is a/(1 – r).

If r ≤ –1 or r > 1, the sequence {rⁿ} is divergent and so, by Equation 3, lim_{n →} s_n does not exist.

Therefore the geometric series diverges in those cases.

cont’d

Series

We summarize the results of Example 2 as follows.

Example 9

Show that the harmonic series

is divergent.

Solution:

For this particular series it’s convenient to consider the partial sums s₂, s₄, s₈, s₁₆, s₃₂, . . . and show that they become large.

Example 9 – Solution

Example 9 – Solution

Similarly, s₃₂ > 1 + , s₆₄ > 1 + , and in general

This shows that → as n → and so {s_n} is divergent.

Therefore the harmonic series diverges.

cont’d

Series

The converse of Theorem 6 is not true in general.

If lim_n→ a_n = 0, we cannot conclude that Σ a_n is convergent.

Series

The Test for Divergence follows from Theorem 6 because, if the series is not divergent, then it is convergent, and so lim_{n →} a_n = 0.

Series