These values substituted in the formula (A), since n is even,

2. Required the first term of the third order of differences of the series 1, 24, 34, 44, &c.

Ans. 60.

3. Required the first term of the fourth order of differences of the series 1, 6, 20, 50, 105, &c.

(193.) To find the nth term of the series

a1; αz; αз; α4; α5; &c.,

we proceed as follows:

Ans. 2.

From the first of the equations (6), of last article, we ob

this value of a substituted in the second of equations (6), gives

a3=a1+2D1+D2,

proceeding in this way we have the following:

a1=a1

a2 = a1+D1

a3=a1+2D1+D2

a1 =a1+3D1+3D2+D3

2

(8)

1. Required the tenth term of the series

1, 4, 8, 13, 19, &c..

3

a1=1; D1=3; D2=1; D ̧=0; and n=10, which values being substituted in (C), we find

9.8

a10=1+9.3+

=64, for the tenth term required.

2

2. Required the nth term of the series 2, 6, 12, 20, 30, &c.

3. What is the nth term of the series

1, 3, 6, 10, 15, 21, &c.?

(194.) To find the sum of n terms of the series

а 1; a 2; aз; as; α5; &c.,

we operate as follows:

Take the new series

2

4

0; a1; a1+a2; a1+a2+ɑ3; a1+a2+aз+a1; &c. (10) Subtracting each term from its next succeeding term, we

which is the same as the original series; hence, the n+1 difference of the series (10), is the same as the n difference of the proposed series, therefore, if in the formula (C), we change a1 into 0, n into n+1, D1 into a1, D2 into D1, D3 into D2, &c., we shall have

1

which expresses the n+1th term of the series (10), but the n+1th term of the series (10) is the same as the sum of n terms of the series

putting this sum equal to S, we shall have for the sum of n terms of the above series, the following expression:

+

D2+&c.

EXAMPLES.

1. Required the sum of n terms of the series

1, 3, 5, 7, 9, &c.

a1 =1, 3, 5

D1 =2, 2

D2 =0.

These values substituted in (D), give

Sn=n+n(n-1)=n2, for the sum of n terms sought.

2. Required the sum of n terms of the series 1, 3, 6, 10, 15, &c.

3. What is the sum of n terms of the series

4. What is the sum of n terms of the series

5. What is the sum of n terms of the series

The answer of the fifth example, being the square of the answer to the fourth example, it follows, that

{1+2+3+4+5+...n}2=13+23+33+43+53+...n3.

SUMMATION OF INFINITE SERIES.

(195.) An Infinite Series is a progression of quantities continued to an infinite number of terms, usually according to some regular law.

(196.) If each term of an infinite series is greater than its preceding term, the series is diverging.

(197.) In general, when each term is less than its preceding, the series is converging, but this is not always the case;

1 1 1

for instance, the series 1++++&c., which is called a 2 3 4

harmonic series, is not a converging series notwithstanding each term is less than its preceding one, its sum to infinity is itself infinite.

(198.) A neutral series is one whose terms are all equal and their signs alternately + and —, thus,

(199.). An ascending series is one in which the powers of the unknown quantities ascend, as

a+bx+cx2+dx3+&c.

(200.) A descending series is one in which the powers of the unknown quantity descend, as

d