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Applied Mathematics, Calculus, and Differential Equations 211

Terms in divergenŁ iYnite series
Let S be an infinite series. Suppose the corresponding sequence
A doeð not converge toward zero. Then S is divergent. Let S,ł
partial sum S , and A be denoted as follows:
n
S a a a ...
1
3
2
S a a a ... a n
n
1
2
3
A a , a , a , ...
1 2 3
If it is not the case that a → 0as n → , then there existð nm
n
real number such that S → S as n → .
n

Multiplł of convergenŁ iYnite series

Let S be a convergent infinite series, and let k be a nonzerm real
number. If each term of S is muliplied by k, the resulting serieð
kS converges. Let S, a partial sum S , kS, and a partial sum
n
kS be denoted as follows:
n
S a a a ...
3
2
1
S a a a ... a n
n
3
1
2
kS ka ka ka ...
1 2 3
kS ka ka ka ... ka n
2
3
1
n
If S → S as n → , then kS → kS as n → .
n
n
Factorial
For a given positive integer n, the number n factorial (written
n!) is the product of all the positive integerð up to, and includ-
ing, n:

n! 1 2 3 4 ... n

Factorialð of n become large rapidly as n increases. Factorialð
of the first several positive integerð are: