Mathematical Formulas for Industrial and Mechanical Engineering

General (nth) term test (also known as the divergence test):

If lim

n!N

a

n

6¼ 0, then the series

P

N

n51

a

n

diverges.

Note This test is a test for divergence only, and says nothing about convergence.

Geometric Series Test

A geometric series has the form

P

N

n50

ar

n

, where “a” is some fixed scalar

(real number). A series of this type will converge provided that jrj, 1, and the

sum is a/(1 2 r). A proof of this result follows.

Consider the kth partial sum, and “r” times the kth partial sum of the series

S

k

5 a 1 ar

1

1 ar

2

1 ar

3

1 ? 1 ar

k

rS

k

5 ar

1

1 ar

2

1 ar

3

1 ? 1 ar

k

1 ar

k11

The difference between rS

k

and S

k

is ðr 2 1ÞS

k

5 aðr

k11

2 1Þ.

Provided that r 6¼1, we can divide by (