13. And if it were required to find the sum of n terms

1

1

+ 十 + &c. 1.2.3 2.3.4 3.4.5 4.5.6

Let z=

1

Then z-- =

2 1.2

+ +
2.3 3.4

-&c. to

'n(n+1)'

1

(n+1)(n+2)

2 (n+1)(n+2) 1.2.3 2.3.4

to n terms, by subtraction.

1

Whence 4-2 (n+1)(n+2) 1.2.3 2.34 3.4.5

terms, by division.

1

1

And consequently+3.4+3.4.5 &c. continued to

1

1

n terms

4 2.(n+1)(n+2)

sum required.

14. Required the sum (s) of the series

2

2

2

+

&c.

3.4.5

Then ·= x( 1 −x+x2 - x3+x+&c.)

1+x

And z=(1+x)x(x-x2+x3-x1+x3 &c.
Whence, by multiplication,

Therefore zx, and x-x2 +x3-x2+x5 &c. =

2

sum required.

1 2 3 4

4 8 16 32

15. Required the sum of the series+++ -+&c.

4 8 16

Let x and s
Let x=2

Then

-=x+2x2+3x3+4x++5x5 &c.

And 2 (1-x) ×(x+2x2+3x3+4x4+5x5 &c.)

Whence, by multiplication,

x+2x2+3x3+ 4x4 &c.

1-2x+x2

16. It is required to find the sum (s) of the series +

(1-x)3

3

=x+4x2+9x+16x1+25x5 &c..

And 2 (1-x)3x(x+4x2+9x3+16x4&c.)=x+x2,

as will be found by actual multiplication.

a a+d
+

mr

17. Required the sum (s) of the series + m

(1-x)a

a+(a+d)x+(a+2d)x2+(a+3d)x3+(a+4d)x+ &c.

And z=(1-x)X {a +(a+d)x+(a+2d) x2+(a+3d)x3 &c. }=(1-x)a+dx,

as will appear by actually multiplying by (1-x)3

Therefore z=(1-x)a+dx; and consequently

= {a(r = 1) + d}

(r-1)a

series required.

α

+

in

sum of the infinite

EXAMPLES.

1. Required the sum of 100 terms of the series 2, 5, 8, 11, 14, &c.

Ans. 15050. 2. Required the sum of 50 terms of the series 1+22 Ans. 42925. +3+4+52 &c. S. It is required to find the sum of the series 1+3x+

8x2+10x+15x4, continued ad infinitum, &c. when x is less than 1.

1

Ans.

(1-x)

4. It is required to find the sum of the series 1+4x +10x2+20x3+35x4, &c. continued ad infinitum, when x

is less than 1.

1

Ans. (1—x)a

5. It is required to find the sum of the infinite series

1

+

1.3

+ +. 3.5 57 7.9

&c.

5

Ans.

1 or

10'

6. Required the sum of 40 terms of the series (1X2) +(3×4)+(5×6)+(7×8) &c.

Ans. 86884 2x-1

7. Required the sum of n terms of the series

8. Required the sum of the infinite series

1 1 1

9. Required the sum of the series

+-+ ·+. +
4 10 20 35'

10. It is required to find the sum of the n terms of series 1+8x+27x2+64x3+129x4, &c. continued ad infinitum.

11. Required the sum of n terms of the series -+-+

6

14. Required the sum of the series +: +

6

6

2.7 7.12 12.17

*The symbol Σ, made use of in these, and some of the following series denotes the sum of an infinite number of terms, and S the sum of n terms.