Series and Partial Sums

Given a sequence \( a_n \), the associated series \( s_n \) is the sum of its first n terms: $$ s_n = a_1+a_2+...+a_n $$ In compact form, this can be written using summation notation as $$ s_n = \sum_{i=1}^n a_i $$

You can think of a series as a running total: each new term adds to everything that came before.

The quantity \( s_n \) is called the n-th partial sum of the series. It tells you the value of the series after adding the first n terms.

More generally, the terms \( s_k \) are called partial sums. Each one represents the sum of the first k terms of the sequence.

$$ s_k = \sum_{i=1}^k a_i = a_1 + a_2 + a_3 + ... + a_k $$

If the sequence \( a_k \) has k terms, then there are exactly k corresponding partial sums:

$$ s_1, s_2, ... , s_k $$

In most cases, the summation starts at \( i=1 \), but this is not required. The index can begin at a different value, depending on how the sequence is defined.

Note. Even when the summation starts at a value different from one (\( i \ne 1 \)), the partial sums are still indexed starting from one. For example, if the summation begins at \( i=2 \) $$ s_n = \sum_{i=2} a_i $$ then the first partial sum is $$ s_1 = a_2 $$ and the next ones are $$ s_2 = a_2+a_3 \\ s_3 = a_2+a_3+a_4 \\ \vdots $$

An Example of a Series

Consider the sequence a_n:

$$ a_n = 2n $$

The first n terms of the sequence a_n are:

$$ 2, 4, 6, 8, 10, ... $$

The corresponding partial sums of the series s_n are:

$$ s_1 = 2 $$

$$ s_2 = 2+4 = 6 $$

$$ s_3 = 2+4+6 = 12 $$

$$ s_4 = 2+4+6+8 = 20 $$

$$ s_5 = 2+4+6+8+10 = 30 $$

So the series s_n of partial sums becomes:

$$ s_1, s_2, s_3, s_4, s_5, ... , s_n $$

$$ 2, 6, 12, 20, 30, ... , s_n $$

The graph below shows the series (in red) plotted on a Cartesian plane.

What’s the difference between a series and a sequence? A sequence is an ordered list of individual terms a_k. A series, on the other hand, is the sequence of partial sums s_k formed by summing the terms of a sequence. For instance, the third term of the sequence is a₃ = 6. The third partial sum of the series is s₃ = a₁ + a₂ + a₃ = 2 + 4 + 6 = 12.

Infinite Series

A series is called an infinite series when n = ∞.

$$ \sum_{k=1}^∞ a_k $$

An infinite series is defined as the limit of the partial sums s_n as n approaches infinity:

$$ \lim_{n \rightarrow ∞} s_n $$

Why is the limit of the series important?

The limit determines the convergence behavior of the series - whether it converges, diverges, or is indeterminate.

Note. A series is considered regular if it either converges or diverges. If it does neither, it's known as an irregular series.

Convergence of a Series

The nature (or behavior) of a series refers to whether the series converges, diverges, or lacks a well-defined limit.

• Convergent series. A series converges if the limit of the partial sums exists and is a finite number. In this case, the sequence s_n tends to a finite value S as n→∞, called the sum of the series: $$ \lim_{n \rightarrow ∞} s_n = S $$

  Note. The sum of the series equals the infinite sum of the terms of the sequence a_n: $$ S = \sum_{k=1}^∞ a_n $$

• Divergent series. A series diverges if the limit of the partial sums tends to positive or negative infinity: $$ \lim_{n \rightarrow ∞} s_n = ±∞ $$
• Indeterminate. A series is indeterminate if the limit does not exist. This is the case for irregular series.

Properties of Infinite Series

Infinite series satisfy the distributive and associative properties, but, in general, they do not satisfy the commutative property.

When working with infinite series, even small changes can make a big difference. In some cases, the sum and the nature of the series remain unchanged. In others, they can change completely.

This is why it is essential to know which properties still apply and which ones break down.

• Distributive property
  If each term of a series is multiplied by a real constant \( c \neq 0 \), the nature of the series (convergent or divergent) does not change. \[ \sum_{n=1}^{\infty} c a_n = c \sum_{n=1}^{\infty} a_n \] In practical terms, if the series \( \sum a_n \) converges, then \( \sum c a_n \) also converges, and its sum is simply scaled by \( c \). If the original series diverges, the new one still diverges.

  Example. Consider the geometric series: \[ \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n \] This series converges and its sum is 2. \[ \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n = 2 \] If each term is multiplied by 3: \[ \sum_{n=0}^{\infty} 3 \left(\frac{1}{2}\right)^n \] the series still converges, and its sum becomes \( 3 \cdot 2 = 6 \). \[ \sum_{n=0}^{\infty} 3 \left(\frac{1}{2}\right)^n = 3 \cdot \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n = 3 \cdot 2 = 6 \]

• Associative property
  Grouping terms into finite blocks does not change the sum. You can combine consecutive terms in different ways, as long as each group contains a finite number of terms. This remains true for both convergent and divergent series.

  Example. Consider a convergent series: \[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \] You can group the terms like this: \[ (1 + \tfrac{1}{2}) + (\tfrac{1}{4} + \tfrac{1}{8}) + \cdots \] or like this: \[ 1 + (\tfrac{1}{2} + \tfrac{1}{4}) + (\tfrac{1}{8} + \cdots) \] In both cases, the series remains convergent and the sum stays the same.

• Failure of the commutative property
  Unlike finite sums, you cannot freely rearrange the terms of an infinite series. Changing the order can alter the sum, or even turn a convergent series into a divergent one. This is especially important for series that are not absolutely convergent.

  Example. Consider the alternating harmonic series: \[ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} \cdots \] This series converges. However, if you rearrange the terms, for example by placing two positive terms followed by one negative term, you can obtain a different sum or even divergence. \[ 1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} \cdots \] This phenomenon is captured by the Riemann rearrangement theorem. It is worth noting that if a series is absolutely convergent, then rearranging its terms does not change the sum.

In conclusion, infinite series do not behave like finite sums.

With infinite series, it is not only what you add that matters, but also the order in which you add it.

And so on.