n terms) =n: and the sum of the latter, +0+1+2+3+, &c.

Because α + a+d+a+2d + a + 3d+,

And a+nd-d+a+nd−2d+ a+nd−3d+ a+nd—4d+, &c. ......+a=s,

The sum

of both, }2a+nd―d+2a+nd—d+2a+nd―d+2a+nd—d+,

PROB. 5. To find the sum of n terms of the series 1, x, x2, x3, &c.

Let 1+x+x2+x3 +, &c. (to xa−1)=s; multiply this series by x, and x+x2+x3 +x1+, &c. (to x")=sx; subtracting the upx"-1 per from the lower, we have -1+x"=sx-s; whence s=

the sum required.

When x is a proper fraction, the sum of the series in infinitum may be found in the same manner.

Thus 1+x+x2 + x3 +, &c.=s.

And x+x3 +x3+x*+,&c.=sx; whence, subtracting as be

1

fore, -1=sx-s, and s= the sum of the series in infinitum. , 1

PROB. 6. To find the sum of an infinite number of terms of the circulating decimal .99999, &c.

PROB. 7. To find the sum of n terms of the series a2+a+d2

+a+2d2+a+3α)*+, &c.

First, by actually squaring the terms, we have

PROB. 8. To find the sum of the infinite series 1+

1

·+ +

3

Whence 1= and therefore s=2, the sum required.

2

و

PROB. 9. To find the sum of n terms of the above series.

Let z=

+, &c. to.

1

n

+ ·+ + +, &c. to

3t 4 5

n

1

PROB. 10. To find the sum s of the infinite series

+, &c.

Let x=

then will x+x2 + x3 + x2 + x3 +, &c.=s;

Substitute :=(s= ) x+x2+x3+x*+x3+, &c.

1-x

Then will z=1—x.x + x2 + x3 +x*+x +, &c. which quantity, by actual multiplication, comes out =x, that is, x=z; and therefore, substituting x for z in the second step, it becomes x+x2+x3

+x2+x2=・ =s; in which, by restoring the value of x, we

have

1 -x

1 1 1 1 1

。 + + + + +, &c. (:
(=———)=1=s,
-)=1=s, the sum re-

2 4 8 16 32

quired.

PROB. 11. To find the sum of 1000 terms of the series 1+ 5+9+13+17+, &c. Ans. 1999000.

PROB. 12. To find the sum of 20 terms of the series 1+3+ 9+27+81 +, &c. Ans. 174339220.

PROB. 13. To find the sum of 12 terms of the series 4+9+ 16+25+, &c. Ans. 1562.

3

PROB. 14. To find the sum of n terms of the series a3+a+di3

PROB. 15. To find the sum of n terms of the series 1+3+ 7+15+31+, &c. Ans. 2"+1−2+n.

PROB. 16. Required the sum of the infinite series.

1

+

2

4

2

PROB. 17. To find the sum of the infinite series + +

3

2 4 8

finitum. Ans. 14.

PROB. 19. To find the sum of the infinite series

+ 1.2.3 2.3.4

PROB. 20. To find the sum of n terms of the above series.

1

PROB. 21. To find the sum of the infinite series

PROB. 22. To find the sum of n terms of the above series.

Let there be given a=N, in which expression x is the logarithm of a*; it is required to find the value of x, that is, the logarithm of (a) the number N.

Let a=1+b, and N=1+n; then will 1+b=1+n, from which, extracting the yth root, we obtain 1+b2 = 1 + n },

L

.b+.
y y

b2
1. +

1 1

n?
·1.-
+

1 1

y y

y y

y

+, &c. 2.3

Here, if y be assumed indefinitely great, the quantities