A.1 Summation formulas and properties

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## A.1 Summation formulas and properties

Given a sequence a1, a2, ... of numbers, the finite sum a1 + a2 + ··· + an, where n is an nonnegative integer, can be written

If n = 0, the value of the summation is defined to be 0. The value of a finite series is always well defined, and its terms can be added in any order.

Given a sequence a1, a2, ... of numbers, the infinite sum a1 + a2 + ··· can be written

which is interpreted to mean

If the limit does not exist, the series diverges; otherwise, it converges. The terms of a convergent series cannot always be added in any order. We can, however, rearrange the terms of an absolutely convergent series, that is, a series for which the series also converges.

#### Linearity

For any real number c and any finite sequences a1, a2, ..., an and b1, b2, ..., bn,

The linearity property is also obeyed by infinite convergent series.

The linearity property can be exploited to manipulate summations incorporating asymptotic notation. For example,

In this equation, the Θ-notation on the left-hand side applies to the variable k, but on the right-hand side, it applies to n. Such manipulations can also be applied to infinite convergent series.

#### Arithmetic series

The summation

is an arithmetic series and has the value

(A.1) |

(A.2) |

#### Sums of squares and cubes

We have the following summations of squares and cubes:

(A.3) |

(A.4) |

#### Geometric series

For real x ≠ 1, the summation

is a geometric or exponential series and has the value

(A.5) |

When the summation is infinite and |x| < 1, we have the infinite decreasing geometric series

(A.6) |

#### Harmonic series

For positive integers n, the nth harmonic number is

(A.7) |

(We shall prove this bound in Section A.2.)

#### Integrating and differentiating series

Additional formulas can be obtained by integrating or differentiating the formulas above. For example, by differentiating both sides of the infinite geometric series (A.6) and multiplying by x, we get

(A.8) |

for |x| < 1.

#### Telescoping series

For any sequence a0, a1, ..., an,

(A.9) |

since each of the terms a1, a2, ..., an-1 is added in exactly once and subtracted out exactly once. We say that the sum telescopes. Similarly,

As an example of a telescoping sum, consider the series

Since we can rewrite each term as

we get

#### Products

The finite product a1 a2 · · an can be written

If n = 0, the value of the product is defined to be 1. We can convert a formula with a product to a formula with a summation by using the identity

Exercises A.1-1

Find a simple formula for

Exercises A.1-2: ⋆

Show that by manipulating the harmonic series.

Exercises A.1-3

Show that for 0 < |x| < 1.

Exercises A.1-4: ⋆

Show that .

Exercises A.1-5: ⋆

Evaluate the sum .

Exercises A.1-6

Prove that by using the linearity property of summations.

Exercises A.1-7

Evaluate the product .

Exercises A.1-8: ⋆

Evaluate the product .

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